- The document is from a Calculus I class at New York University and covers evaluating definite integrals.
- It discusses using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval.
- Examples are provided of using the midpoint rule to estimate a definite integral, and properties of definite integrals like additivity and comparison properties are reviewed.
This document outlines a calculus lecture on evaluating definite integrals. The lecture will:
1) Use the Evaluation Theorem to evaluate definite integrals and write antiderivatives as indefinite integrals.
2) Interpret definite integrals as the "net change" of a function over an interval.
3) Provide examples of evaluating definite integrals and estimating integrals using the midpoint rule.
4) Discuss properties of integrals such as additivity and illustrate how a definite integral from a to c can be broken into integrals from a to b and b to c.
This document contains notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The section discusses using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. It provides examples of evaluating definite integrals and computing area with integrals using the Second Fundamental Theorem of Calculus.
The document discusses statistical concepts including Gaussian distributions, standard deviation, confidence intervals, t-tests, and calibration curves. It provides examples of how to calculate the mean, standard deviation, confidence intervals using t-tables, and how to perform t-tests to compare two data sets. It also describes constructing a calibration curve using the method of least squares to determine the best-fit line and using that line to find the concentration of an unknown sample.
This document discusses a study analyzing the coronas (gas discharge visualizations) of apple tree leaves and fruits using the GDV Assistant system. The researchers recorded coronas under different conditions to analyze plant vitality and stress levels. They used various machine learning algorithms to analyze the parameterized corona images. The results showed coronas provide useful information about plant stress and variety. However, they could not differentiate between organically and conventionally grown fruit that were similar in standard quality measures. The document describes the GDV Assistant system parameters, recording methodology, classification problems analyzed, machine learning methods used, and results.
05210401 P R O B A B I L I T Y T H E O R Y A N D S T O C H A S T I C P R...guestd436758
This document appears to be an exam for a Probability Theory and Stochastic Process course, consisting of 8 questions across 4 pages. It covers topics like events, probability, random variables, probability distributions, moments, central limit theorem, stationary processes, power spectral density, and linear time-invariant systems. Students are instructed to answer any 5 of the 8 questions, which include problems calculating probabilities, distributions, moments, variances, correlation coefficients, power spectral densities, and network responses. Diagrams are provided for reference.
This document proposes an approximate Bayesian inference method for estimating propensity scores under nonresponse. It involves treating the estimating equations as random variables and assigning a prior distribution to the transformed parameters. Samples are drawn from the posterior distribution of the parameters given the observed data to make inferences. The method is shown to be asymptotically consistent and confidence regions can be constructed from the posterior samples. Extensions are discussed to incorporate auxiliary variables and perform Bayesian model selection by assigning a spike-and-slab prior over the model parameters.
1.differential approach to cardioid distribution -1-6Alexander Decker
This document presents a differential equation approach to deriving the probability density function of the cardioid distribution. It shows that the cardioid distribution can be obtained as the solution to a second-order non-homogeneous linear differential equation under certain initial conditions. It also describes how the Mobius transformation can be used to map the cardioid distribution on a circle to real-valued Cauchy-type distributions. Specifically, it presents a Mobius transformation that maps points on the unit circle to the real line, allowing the derivation of new unimodal symmetric distributions from the original cardioid distribution.
11.0001www.iiste.org call for paper.differential approach to cardioid distrib...Alexander Decker
This document summarizes a research paper that derives the probability density function (pdf) of the cardioid distribution through solving a differential equation. It then uses a Mobius transformation to map the cardioid distribution on a circle to new symmetric and unimodal distributions on the real line, called Cauchy type models. Graphs of the Cauchy type model pdfs are provided for different parameter values.
This document outlines a calculus lecture on evaluating definite integrals. The lecture will:
1) Use the Evaluation Theorem to evaluate definite integrals and write antiderivatives as indefinite integrals.
2) Interpret definite integrals as the "net change" of a function over an interval.
3) Provide examples of evaluating definite integrals and estimating integrals using the midpoint rule.
4) Discuss properties of integrals such as additivity and illustrate how a definite integral from a to c can be broken into integrals from a to b and b to c.
This document contains notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The section discusses using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. It provides examples of evaluating definite integrals and computing area with integrals using the Second Fundamental Theorem of Calculus.
The document discusses statistical concepts including Gaussian distributions, standard deviation, confidence intervals, t-tests, and calibration curves. It provides examples of how to calculate the mean, standard deviation, confidence intervals using t-tables, and how to perform t-tests to compare two data sets. It also describes constructing a calibration curve using the method of least squares to determine the best-fit line and using that line to find the concentration of an unknown sample.
This document discusses a study analyzing the coronas (gas discharge visualizations) of apple tree leaves and fruits using the GDV Assistant system. The researchers recorded coronas under different conditions to analyze plant vitality and stress levels. They used various machine learning algorithms to analyze the parameterized corona images. The results showed coronas provide useful information about plant stress and variety. However, they could not differentiate between organically and conventionally grown fruit that were similar in standard quality measures. The document describes the GDV Assistant system parameters, recording methodology, classification problems analyzed, machine learning methods used, and results.
05210401 P R O B A B I L I T Y T H E O R Y A N D S T O C H A S T I C P R...guestd436758
This document appears to be an exam for a Probability Theory and Stochastic Process course, consisting of 8 questions across 4 pages. It covers topics like events, probability, random variables, probability distributions, moments, central limit theorem, stationary processes, power spectral density, and linear time-invariant systems. Students are instructed to answer any 5 of the 8 questions, which include problems calculating probabilities, distributions, moments, variances, correlation coefficients, power spectral densities, and network responses. Diagrams are provided for reference.
This document proposes an approximate Bayesian inference method for estimating propensity scores under nonresponse. It involves treating the estimating equations as random variables and assigning a prior distribution to the transformed parameters. Samples are drawn from the posterior distribution of the parameters given the observed data to make inferences. The method is shown to be asymptotically consistent and confidence regions can be constructed from the posterior samples. Extensions are discussed to incorporate auxiliary variables and perform Bayesian model selection by assigning a spike-and-slab prior over the model parameters.
1.differential approach to cardioid distribution -1-6Alexander Decker
This document presents a differential equation approach to deriving the probability density function of the cardioid distribution. It shows that the cardioid distribution can be obtained as the solution to a second-order non-homogeneous linear differential equation under certain initial conditions. It also describes how the Mobius transformation can be used to map the cardioid distribution on a circle to real-valued Cauchy-type distributions. Specifically, it presents a Mobius transformation that maps points on the unit circle to the real line, allowing the derivation of new unimodal symmetric distributions from the original cardioid distribution.
11.0001www.iiste.org call for paper.differential approach to cardioid distrib...Alexander Decker
This document summarizes a research paper that derives the probability density function (pdf) of the cardioid distribution through solving a differential equation. It then uses a Mobius transformation to map the cardioid distribution on a circle to new symmetric and unimodal distributions on the real line, called Cauchy type models. Graphs of the Cauchy type model pdfs are provided for different parameter values.
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
This document summarizes key concepts from Chapter 2 of a book on statistical methods for handling incomplete data. It introduces the likelihood-based approach and defines key terms like the likelihood function, maximum likelihood estimator, Fisher information, and missing at random. The chapter also provides examples of observed likelihood functions for censored regression and survival analysis models with missing data.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and to use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the Extreme Value Theorem, which states that a continuous function on a closed interval attains maximum and minimum values. Examples are given to show the importance of the hypotheses in the theorem.
Expectation Maximization Algorithm with Combinatorial AssumptionLoc Nguyen
Expectation maximization (EM) algorithm is a popular and powerful mathematical method for parameter estimation in case that there exist both observed data and hidden data. The EM process depends on an implicit relationship between observed data and hidden data which is specified by a mapping function in traditional EM and a joint probability density function (PDF) in practical EM. However, the mapping function is vague and impractical whereas the joint PDF is not easy to be defined because of heterogeneity between observed data and hidden data. The research aims to improve competency of EM by making it more feasible and easier to be specified, which removes the vagueness. Therefore, the research proposes an assumption that observed data is the combination of hidden data which is realized as an analytic function where data points are numerical. In other words, observed points are supposedly calculated from hidden points via regression model. Mathematical computations and proofs indicate feasibility and clearness of the proposed method which can be considered as an extension of EM.
Some Generalized I nformation Inequalitiesijitjournal
Information inequalities are
very useful and play a fundamental role in the literature of Information
Theory. Applicati
ons of information inequalities
have discussed by well
-
kn
own authors like as Dragomir,
Taneja
and many researchers etc. In this research paper, we shall consider some
new functional
information inequalities in the form of
generalized
information divergence measures. We shall also
consider relations between Csiszar’s f
-
divergence, new f
-
divergence and other well
-
known divergence
measures using information inequalities.
Numerical bounds of information divergence measure have also
studied
This document discusses predictive mean matching (PMM) imputation in survey sampling. It begins with an outline and overview of the basic setup, assumptions, and PMM imputation method. It then presents three main theorems: 1) the asymptotic normality of the PMM estimator when the regression parameter β* is known, 2) the asymptotic normality when β* is estimated, and 3) the asymptotic properties of nearest neighbor imputation. The document also discusses variance estimation for the PMM estimator using replication methods like the bootstrap or jackknife. In summary, it provides a theoretical analysis of the asymptotic properties of PMM imputation and approaches for estimating the variance.
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
ECE 302 Spring 2012 covers practice problems involving various continuous and discrete random variables, including uniform, exponential, normal, lognormal, Rayleigh, Cauchy, Pareto, Gaussian mixture, Erlang, and Laplace distributions. The document provides an example problem solving a uniform random variable. It also lists some suggested reading materials and references textbooks for further information.
This 3-sentence summary provides the essential information about the document:
The document contains a mathematics quiz with 5 questions about matrix operations such as matrix multiplication, finding submatrices, matrix transposition, and combining matrices using augmentation and stacking. Sample matrices are provided with each question and the solutions to each question are worked out step-by-step showing the calculations to arrive at the final answers.
11 X1 T05 01 division of an interval (2010)Nigel Simmons
Here are the steps to solve this problem:
1) Write down the endpoints of the interval: (-3,4) and (5,6)
2) The ratio given is 1:3
3) Let the interval from the first endpoint to P be 1 unit
4) Then the interval from P to the second endpoint is 3 units
5) Use the ratio of distances formula:
xP - x1 / x2 - x1 = 1/(1+3)
yP - y1 / y2 - y1 = 1/(1+3)
6) Solve the two equations simultaneously to find the coordinates of P.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
Generalization of Compositons of Cellular Automata on GroupsYoshihiro Mizoguchi
The document discusses cellular automata (CA) and proposes generalizing CA theory. It defines CA on groups where the cell space is a group rather than a lattice. This allows local functions to have different neighborhood domains. Operations are introduced to combine CA with different neighborhood types. Decomposing complex CA into simpler components using these operations is explored, as is studying how operations relate to CA dynamics. The goal is a more generalized CA theory.
Lesson 25: The Fundamental Theorem of Calculus (handout)Matthew Leingang
This document outlines a calculus lecture on the fundamental theorem of calculus. It discusses defining an area function from an integral, proves the first fundamental theorem of calculus, and gives examples of how differentiation and integration are reverse processes. It also provides brief biographies of several important mathematicians like Newton and Leibniz related to the development of calculus. The lecture concludes with an example of differentiating a function defined by an integral using the first fundamental theorem.
The document discusses classification algorithms in machine learning. It introduces classification problems using the Iris flower dataset as an example, which contains measurements of Iris flowers to classify them into three species. It then discusses two classic classification algorithms - logistic regression and Gaussian discriminant analysis. Logistic regression uses a sigmoid function to generate predictions, while Gaussian discriminant analysis assumes a Gaussian distribution of the data. The document also demonstrates an application of these algorithms to classify handwritten digits.
Sixth order hybrid block method for the numerical solution of first order ini...Alexander Decker
This document presents a sixth-order hybrid block method for numerically solving first order initial value problems. The method is derived by combining five consistent finite difference schemes obtained from multistep collocation of the differential system and interpolation at grid and off-grid points. The accuracy of the method is demonstrated on some standard test problems.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
The document is about calculating areas and distances using calculus. It discusses calculating areas of rectangles, parallelograms, triangles and other polygons. For curved regions, it discusses Archimedes' method of approximating areas using inscribed polygons and letting the number of sides approach infinity. It also discusses calculating distances traveled as the limit of approximating distances over smaller time intervals. The objectives are to compute areas using approximating rectangles and distances using approximating time intervals.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Having explored the area problem for curved regions or regions below graphs of functions, we define the definite integral and state some of its properties. It's defined for functions which are continuous or at worst have finitely many jump or removable discontinuities. It's "linear" with respect to addition and scaling of functions. And it preserves order between functions.
This slideshow document provides integration rules and their corresponding integrals. It contains the function to integrate on one slide followed by the integral solution on the next slide, covering multiple integration rules important for the course. The document serves as a reference for learning integration through examples of different functions and seeing the resulting integrals.
The document describes angles and their measurement in radians and degrees. It defines an angle, measures angles in radians using central angles of a circle, and defines the relationship between radians and degrees. Examples show converting between radians and degrees, finding coterminal angles, complementary and supplementary angles, and calculating arc length and linear speed using angular measurements.
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
This document summarizes key concepts from Chapter 2 of a book on statistical methods for handling incomplete data. It introduces the likelihood-based approach and defines key terms like the likelihood function, maximum likelihood estimator, Fisher information, and missing at random. The chapter also provides examples of observed likelihood functions for censored regression and survival analysis models with missing data.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and to use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the Extreme Value Theorem, which states that a continuous function on a closed interval attains maximum and minimum values. Examples are given to show the importance of the hypotheses in the theorem.
Expectation Maximization Algorithm with Combinatorial AssumptionLoc Nguyen
Expectation maximization (EM) algorithm is a popular and powerful mathematical method for parameter estimation in case that there exist both observed data and hidden data. The EM process depends on an implicit relationship between observed data and hidden data which is specified by a mapping function in traditional EM and a joint probability density function (PDF) in practical EM. However, the mapping function is vague and impractical whereas the joint PDF is not easy to be defined because of heterogeneity between observed data and hidden data. The research aims to improve competency of EM by making it more feasible and easier to be specified, which removes the vagueness. Therefore, the research proposes an assumption that observed data is the combination of hidden data which is realized as an analytic function where data points are numerical. In other words, observed points are supposedly calculated from hidden points via regression model. Mathematical computations and proofs indicate feasibility and clearness of the proposed method which can be considered as an extension of EM.
Some Generalized I nformation Inequalitiesijitjournal
Information inequalities are
very useful and play a fundamental role in the literature of Information
Theory. Applicati
ons of information inequalities
have discussed by well
-
kn
own authors like as Dragomir,
Taneja
and many researchers etc. In this research paper, we shall consider some
new functional
information inequalities in the form of
generalized
information divergence measures. We shall also
consider relations between Csiszar’s f
-
divergence, new f
-
divergence and other well
-
known divergence
measures using information inequalities.
Numerical bounds of information divergence measure have also
studied
This document discusses predictive mean matching (PMM) imputation in survey sampling. It begins with an outline and overview of the basic setup, assumptions, and PMM imputation method. It then presents three main theorems: 1) the asymptotic normality of the PMM estimator when the regression parameter β* is known, 2) the asymptotic normality when β* is estimated, and 3) the asymptotic properties of nearest neighbor imputation. The document also discusses variance estimation for the PMM estimator using replication methods like the bootstrap or jackknife. In summary, it provides a theoretical analysis of the asymptotic properties of PMM imputation and approaches for estimating the variance.
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
ECE 302 Spring 2012 covers practice problems involving various continuous and discrete random variables, including uniform, exponential, normal, lognormal, Rayleigh, Cauchy, Pareto, Gaussian mixture, Erlang, and Laplace distributions. The document provides an example problem solving a uniform random variable. It also lists some suggested reading materials and references textbooks for further information.
This 3-sentence summary provides the essential information about the document:
The document contains a mathematics quiz with 5 questions about matrix operations such as matrix multiplication, finding submatrices, matrix transposition, and combining matrices using augmentation and stacking. Sample matrices are provided with each question and the solutions to each question are worked out step-by-step showing the calculations to arrive at the final answers.
11 X1 T05 01 division of an interval (2010)Nigel Simmons
Here are the steps to solve this problem:
1) Write down the endpoints of the interval: (-3,4) and (5,6)
2) The ratio given is 1:3
3) Let the interval from the first endpoint to P be 1 unit
4) Then the interval from P to the second endpoint is 3 units
5) Use the ratio of distances formula:
xP - x1 / x2 - x1 = 1/(1+3)
yP - y1 / y2 - y1 = 1/(1+3)
6) Solve the two equations simultaneously to find the coordinates of P.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
Generalization of Compositons of Cellular Automata on GroupsYoshihiro Mizoguchi
The document discusses cellular automata (CA) and proposes generalizing CA theory. It defines CA on groups where the cell space is a group rather than a lattice. This allows local functions to have different neighborhood domains. Operations are introduced to combine CA with different neighborhood types. Decomposing complex CA into simpler components using these operations is explored, as is studying how operations relate to CA dynamics. The goal is a more generalized CA theory.
Lesson 25: The Fundamental Theorem of Calculus (handout)Matthew Leingang
This document outlines a calculus lecture on the fundamental theorem of calculus. It discusses defining an area function from an integral, proves the first fundamental theorem of calculus, and gives examples of how differentiation and integration are reverse processes. It also provides brief biographies of several important mathematicians like Newton and Leibniz related to the development of calculus. The lecture concludes with an example of differentiating a function defined by an integral using the first fundamental theorem.
The document discusses classification algorithms in machine learning. It introduces classification problems using the Iris flower dataset as an example, which contains measurements of Iris flowers to classify them into three species. It then discusses two classic classification algorithms - logistic regression and Gaussian discriminant analysis. Logistic regression uses a sigmoid function to generate predictions, while Gaussian discriminant analysis assumes a Gaussian distribution of the data. The document also demonstrates an application of these algorithms to classify handwritten digits.
Sixth order hybrid block method for the numerical solution of first order ini...Alexander Decker
This document presents a sixth-order hybrid block method for numerically solving first order initial value problems. The method is derived by combining five consistent finite difference schemes obtained from multistep collocation of the differential system and interpolation at grid and off-grid points. The accuracy of the method is demonstrated on some standard test problems.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
The document is about calculating areas and distances using calculus. It discusses calculating areas of rectangles, parallelograms, triangles and other polygons. For curved regions, it discusses Archimedes' method of approximating areas using inscribed polygons and letting the number of sides approach infinity. It also discusses calculating distances traveled as the limit of approximating distances over smaller time intervals. The objectives are to compute areas using approximating rectangles and distances using approximating time intervals.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Having explored the area problem for curved regions or regions below graphs of functions, we define the definite integral and state some of its properties. It's defined for functions which are continuous or at worst have finitely many jump or removable discontinuities. It's "linear" with respect to addition and scaling of functions. And it preserves order between functions.
This slideshow document provides integration rules and their corresponding integrals. It contains the function to integrate on one slide followed by the integral solution on the next slide, covering multiple integration rules important for the course. The document serves as a reference for learning integration through examples of different functions and seeing the resulting integrals.
The document describes angles and their measurement in radians and degrees. It defines an angle, measures angles in radians using central angles of a circle, and defines the relationship between radians and degrees. Examples show converting between radians and degrees, finding coterminal angles, complementary and supplementary angles, and calculating arc length and linear speed using angular measurements.
The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will cover fundamentals of calculus including limits, derivatives, integrals, and optimization. It will meet twice a week for lectures and recitations. Assessment will include weekly homework, biweekly quizzes, a midterm exam, and a final exam. Grades will be calculated based on scores on these assessments. The required textbook can be purchased in hardcover, looseleaf, or online formats. Students are encouraged to contact the professor or TAs with any questions.
This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.
This document summarizes a calculus lecture on the Mean Value Theorem. It begins with announcements about exams and assignments. It then outlines the topics to be covered: Rolle's Theorem and the Mean Value Theorem, applications of the MVT, and why the MVT is important. It provides heuristic motivations and mathematical statements of Rolle's Theorem and the Mean Value Theorem. It also includes a proof of the Mean Value Theorem using Rolle's Theorem.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
- The document is a section from a calculus course at NYU that discusses using derivatives to determine the shapes of curves.
- It covers using the first derivative to determine if a function is increasing or decreasing over an interval using the Increasing/Decreasing Test. It also discusses using the second derivative to determine if a function is concave up or down over an interval using the Second Derivative Test.
- Examples are provided to demonstrate finding intervals of monotonicity for functions and classifying critical points as local maxima, minima or neither using the First Derivative Test.
Lesson 17: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document appears to be a lecture on indeterminate forms and L'Hopital's rule from a Calculus I course at New York University. It includes:
1) An introduction to different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, etc.
2) Examples of limits that are indeterminate forms and experiments calculating their values.
3) A discussion of dividing limits and exceptions where the limit may not exist even when the numerator approaches a finite number and denominator approaches zero.
4) An outline of the topics to be covered, including L'Hopital's rule, other indeterminate limits, and a summary.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
Silberberg Chemistry Molecular Nature Of Matter And Change 4e Copy2jeksespina
There are three main types of radioactive decay: alpha, beta, and gamma. Alpha decay involves emitting an alpha particle (helium nucleus) which decreases the mass and atomic numbers by 4 and 2 respectively. Beta decay involves emitting an electron or positron, which does not change mass number but increases or decreases the atomic number by 1. Gamma decay involves emitting high-energy photons without changing the nucleus. Nuclear equations must balance the total numbers of protons and nucleons between reactants and products.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
The document discusses the definite integral, including computing it using Riemann sums, estimating it using approximations like the midpoint rule, and reasoning about its properties. It outlines the topics to be covered, such as recalling previous concepts and comparing properties of integrals. Formulas are provided for calculating Riemann sums using different representative points within the intervals.
The document provides an overview of curve sketching in calculus. It discusses objectives like sketching a function graph completely by identifying zeros, asymptotes, critical points, maxima/minima and inflection points. Examples are provided to demonstrate the increasing/decreasing test using derivatives and the concavity test to determine concave up/down regions. A step-by-step process is outlined to graph functions which involves analyzing monotonicity using the sign chart of the derivative and concavity using the second derivative sign chart. This is demonstrated on sketching the graph of a cubic function f(x)=2x^3-3x^2-12x.
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Mel Anthony Pepito
The document summarizes key points from Section 5.3 of a calculus class at New York University on evaluating definite integrals. It outlines the objectives of the section, provides an example of using the midpoint rule to estimate a definite integral, and reviews properties and comparison properties of definite integrals, including how to interpret definite integrals as the net change of a function over an interval.
This document outlines lecture material on evaluating definite integrals from a Calculus I course at New York University. The lecture covers using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Properties of the definite integral such as additivity and comparison properties are discussed. The Second Fundamental Theorem of Calculus is proved, relating the definite integral of a function to the antiderivative of that function. Examples are provided of using this theorem to compute areas under curves defined by functions.
The document is a lecture note on the fundamental theorem of calculus from a Calculus I class at New York University. It provides announcements about upcoming exams and assignments. It then outlines the key topics to be covered, including the first fundamental theorem of calculus and how to differentiate functions defined by integrals. Examples are provided to illustrate using integrals to find the area under a curve and how this relates to the derivative of the area function.
The document provides information about an upcoming calculus exam, quiz, and lecture on curve sketching. It outlines the procedure for sketching curves, including using the increasing/decreasing test and concavity test. It then provides an example of sketching a cubic function, showing the steps of finding critical points, inflection points, and putting together a sign chart to determine the function's monotonicity and concavity over intervals.
This document provides an example of graphing a cubic function step-by-step. It begins by finding the derivative to determine where the function is increasing and decreasing. The derivative is factored to yield a sign chart showing the function is decreasing on (-∞,-1) and (2,∞) and increasing on (-1,2). This information will be used to sketch the graph of the cubic function.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Mel Anthony Pepito
The document discusses the fundamental theorem of calculus. It begins by outlining the topics to be covered, including a review of the second fundamental theorem of calculus, an explanation of the first fundamental theorem of calculus, and examples of differentiating functions defined by integrals. It then provides more details on these topics, such as defining the integral as a limit, stating the second fundamental theorem, using integrals to represent concepts like distance traveled and mass, and working through an example of finding the derivative of a function defined by an integral.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Matthew Leingang
g(x) represents the area under the curve of f(t) from 0 to x. As x increases from 0 to 10, g(x) will increase, representing the accumulating area under f(t) over the interval [0,x].
The document provides an overview of curve sketching techniques for functions. It discusses objectives like graphing functions completely and indicating key features. It then covers topics like the increasing/decreasing test, concavity test, a checklist for graphing functions, and examples of graphing cubic, quartic, and other types of functions. Examples are worked through step-by-step to demonstrate the process of analyzing monotonicity, concavity, and asymptotes to fully sketch the graph of various functions.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides an outline and review for the final exam in Math 1a Integration. It covers key topics like the Riemann integral, properties of the integral, comparison properties, the Fundamental Theorem of Calculus, and integration by substitution. Examples are provided to illustrate computing integrals using properties, the relationship between antiderivatives and derivatives defined by integrals, and applying the Fundamental Theorem of Calculus to find derivatives.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Matthew Leingang
The document outlines a calculus class lecture on the fundamental theorem of calculus, including recalling the second fundamental theorem, stating the first fundamental theorem, and providing examples of differentiating functions defined by integrals. It gives announcements for upcoming class sections and exam dates, lists the objectives of the current section, and provides an outline of topics to be covered including area as a function, statements and proofs of the theorems, and applications to differentiation.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Mel Anthony Pepito
The document provides an overview of Section 5.4 on the Fundamental Theorem of Calculus from a Calculus I course at New York University. It outlines topics to be covered, including recalling the Second Fundamental Theorem, stating the First Fundamental Theorem, and differentiating functions defined by integrals. Examples are provided to illustrate using the theorems to find derivatives and integrals.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 handout)Matthew Leingang
This document contains lecture notes on the fundamental theorem of calculus from a Calculus I class. The notes discuss:
1) The first and second fundamental theorems of calculus, which relate differentiation and integration as inverse processes.
2) How to use the first fundamental theorem to differentiate functions defined by integrals.
3) Biographies of several mathematicians involved in the development of calculus, including Newton, Leibniz, Gregory and Barrow.
The document provides an example of using the substitution method to evaluate the indefinite integral ∫(x2 + 3)3 4x dx. It introduces the substitution u = x2 + 3, which allows the integral to be rewritten as ∫u3 2 du and then evaluated as (1/2)u4 = (1/2)(x2 + 3)4. The solution is compared to directly integrating the expanded polynomial. The document outlines the theory and notation of substitution for indefinite integrals.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
The document is a lecture note on derivatives and the shapes of curves. It discusses the mean value theorem and its applications to determining monotonicity and concavity of functions. Specifically, it covers:
- Using the first derivative test to find intervals where a function is increasing or decreasing by determining where the derivative is positive or negative
- Examples of applying this process to functions like x2 - 1 and x2/3(x + 2)
- Definitions of increasing, decreasing, and concavity
- How the second derivative test can determine concavity by examining the sign of the second derivative
The document provides information about an upcoming calculus class, including announcements of an upcoming quiz and that class will be held on November 24. It then outlines the objectives and importance of curve sketching functions by analyzing properties like critical points, maxima/minima, and inflection points. The remainder of the document provides examples and steps for sketching curves of specific cubic and quartic functions through analyzing monotonicity using the first derivative and concavity using the second derivative.
Similar to Lesson 26: Evaluating Definite Integrals (20)
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.
Essentials of Automations: Exploring Attributes & Automation ParametersSafe Software
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Generating privacy-protected synthetic data using Secludy and MilvusZilliz
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Your One-Stop Shop for Python Success: Top 10 US Python Development Providersakankshawande
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In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
This presentation delves into the development of a system designed to mimic Galileo's Open Service signal using software-defined radio (SDR) technology. We'll begin with a foundational overview of both Global Navigation Satellite Systems (GNSS) and the intricacies of digital signal processing.
The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
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The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
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Bram Verhoef, Head of Machine Learning at Axelera AI, presents the “How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-efficient Computer Vision” tutorial at the May 2024 Embedded Vision Summit.
As artificial intelligence inference transitions from cloud environments to edge locations, computer vision applications achieve heightened responsiveness, reliability and privacy. This migration, however, introduces the challenge of operating within the stringent confines of resource constraints typical at the edge, including small form factors, low energy budgets and diminished memory and computational capacities. Axelera AI addresses these challenges through an innovative approach of performing digital computations within memory itself. This technique facilitates the realization of high-performance, energy-efficient and cost-effective computer vision capabilities at the thin and thick edge, extending the frontier of what is achievable with current technologies.
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[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...Jason Yip
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Skybuffer SAM4U tool for SAP license adoptionTatiana Kojar
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Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
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Introduction of Cybersecurity with OSS at Code Europe 2024Hiroshi SHIBATA
I develop the Ruby programming language, RubyGems, and Bundler, which are package managers for Ruby. Today, I will introduce how to enhance the security of your application using open-source software (OSS) examples from Ruby and RubyGems.
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How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
1. Section 5.3
Evaluating Definite Integrals
V63.0121.002.2010Su, Calculus I
New York University
June 21, 2010
Announcements
Final Exam is Thursday in class
2. Announcements
Sections 5.3–5.4 today
Section 5.5 tomorrow
Review and Movie Day
Wednesday
Final exam Thursday
roughly half-and-half
MC/FR
FR is all post-midterm
MC might have some
pre-midterm stuff on it
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 2 / 44
3. Resurrection Policy
If your final score beats your midterm score, we will add 10% to its weight,
and subtract 10% from the midterm weight.
Image credit: Scott Beale / Laughing Squid
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 3 / 44
4. Objectives
Use the Evaluation Theorem
to evaluate definite integrals.
Write antiderivatives as
indefinite integrals.
Interpret definite integrals as
“net change” of a function
over an interval.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 4 / 44
5. Outline
Last time: The Definite Integral
The definite integral as a limit
Properties of the integral
Comparison Properties of the Integral
Evaluating Definite Integrals
Examples
The Integral as Total Change
Indefinite Integrals
My first table of integrals
Computing Area with integrals
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 5 / 44
6. The definite integral as a limit
Definition
If f is a function defined on [a, b], the definite integral of f from a to b
is the number
b n
f (x) dx = lim f (ci ) ∆x
a n→∞
i=1
b−a
where ∆x = , and for each i, xi = a + i∆x, and ci is a point in
n
[xi−1 , xi ].
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite integral
b
f (x) dx exists and is the same for any choice of ci .
a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 6 / 44
7. Notation/Terminology
b
f (x) dx
a
— integral sign (swoopy S)
f (x) — integrand
a and b — limits of integration (a is the lower limit and b the
upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
The process of computing an integral is called integration
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 7 / 44
8. Example
1
4
Estimate dx using the midpoint rule and four divisions.
0 1 + x2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 8 / 44
9. Example
1
4
Estimate dx using the midpoint rule and four divisions.
0 1 + x2
Solution
Dividing up [0, 1] into 4 pieces gives
1 2 3 4
x0 = 0, x1 = , x2 = , x3 = , x4 =
4 4 4 4
So the midpoint rule gives
1 4 4 4 4
M4 = 2
+ 2
+ 2
+
4 1 + (1/8) 1 + (3/8) 1 + (5/8) 1 + (7/8)2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 8 / 44
10. Example
1
4
Estimate dx using the midpoint rule and four divisions.
0 1 + x2
Solution
Dividing up [0, 1] into 4 pieces gives
1 2 3 4
x0 = 0, x1 = , x2 = , x3 = , x4 =
4 4 4 4
So the midpoint rule gives
1 4 4 4 4
M4 = 2
+ 2
+ 2
+
4 1 + (1/8) 1 + (3/8) 1 + (5/8) 1 + (7/8)2
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 8 / 44
12. Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant. Then
b
1. c dx = c(b − a)
a
b b b
2. [f (x) + g (x)] dx = f (x) dx + g (x) dx.
a a a
b b
3. cf (x) dx = c f (x) dx.
a a
b b b
4. [f (x) − g (x)] dx = f (x) dx − g (x) dx.
a a a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 9 / 44
13. More Properties of the Integral
Conventions:
a b
f (x) dx = − f (x) dx
b a
a
f (x) dx = 0
a
This allows us to have
c b c
5. f (x) dx = f (x) dx + f (x) dx for all a, b, and c.
a a b
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 10 / 44
14. Definite Integrals We Know So Far
If the integral computes an
area and we know the area,
we can use that. For
y
instance,
1
π
1 − x 2 dx =
0 2
By brute force we computed x
1 1
1 1
x 2 dx = x 3 dx =
0 3 0 4
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 11 / 44
15. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then
b
f (x) dx ≥ 0
a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 12 / 44
16. Integral of a nonnegative function is nonnegative
Proof.
If f (x) ≥ 0 for all x in [a, b], then for any number of divisions n and choice
of sample points {ci }:
n n
Sn = f (ci ) ∆x ≥ 0 · ∆x = 0
i=1 ≥0 i=1
Since Sn ≥ 0 for all n, the limit of {Sn } is nonnegative, too:
b
f (x) dx = lim Sn ≥ 0
a n→∞
≥0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 13 / 44
17. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then
b
f (x) dx ≥ 0
a
7. If f (x) ≥ g (x) for all x in [a, b], then
b b
f (x) dx ≥ g (x) dx
a a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 14 / 44
18. The definite integral is “increasing”
Proof.
Let h(x) = f (x) − g (x). If f (x) ≥ g (x) for all x in [a, b], then h(x) ≥ 0
for all x in [a, b]. So by the previous property
b
h(x) dx ≥ 0
a
This means that
b b b b
f (x) dx − g (x) dx = (f (x) − g (x)) dx = h(x) dx ≥ 0
a a a a
So
b b
f (x) dx ≥ g (x) dx
a a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 15 / 44
19. Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f (x) ≥ 0 for all x in [a, b], then
b
f (x) dx ≥ 0
a
7. If f (x) ≥ g (x) for all x in [a, b], then
b b
f (x) dx ≥ g (x) dx
a a
8. If m ≤ f (x) ≤ M for all x in [a, b], then
b
m(b − a) ≤ f (x) dx ≤ M(b − a)
a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 16 / 44
20. Bounding the integral using bounds of the function
Proof.
If m ≤ f (x) ≤ M on for all x in [a, b], then by the previous property
b b b
m dx ≤ f (x) dx ≤ M dx
a a a
By Property 1, the integral of a constant function is the product of the
constant and the width of the interval. So:
b
m(b − a) ≤ f (x) dx ≤ M(b − a)
a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 17 / 44
21. Estimating an integral with inequalities
Example
2
1
Estimate dx using Property 8.
1 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 18 / 44
22. Estimating an integral with inequalities
Example
2
1
Estimate dx using Property 8.
1 x
Solution
Since
1 1 1
1 ≤ x ≤ 2 =⇒ ≤ ≤
2 x 1
we have
2
1 1
· (2 − 1) ≤ dx ≤ 1 · (2 − 1)
2 1 x
or
2
1 1
≤ dx ≤ 1
2 1 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 18 / 44
23. Outline
Last time: The Definite Integral
The definite integral as a limit
Properties of the integral
Comparison Properties of the Integral
Evaluating Definite Integrals
Examples
The Integral as Total Change
Indefinite Integrals
My first table of integrals
Computing Area with integrals
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 19 / 44
24. Socratic proof
The definite integral of
velocity measures
displacement (net distance)
The derivative of
displacement is velocity
So we can compute
displacement with the
definite integral or the
antiderivative of velocity
But any function can be a
velocity function, so . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 20 / 44
25. Theorem of the Day
Theorem (The Second Fundamental Theorem of Calculus)
Suppose f is integrable on [a, b] and f = F for another function F , then
b
f (x) dx = F (b) − F (a).
a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 21 / 44
26. Theorem of the Day
Theorem (The Second Fundamental Theorem of Calculus)
Suppose f is integrable on [a, b] and f = F for another function F , then
b
f (x) dx = F (b) − F (a).
a
Note
In Section 5.3, this theorem is called “The Evaluation Theorem”. Nobody
else in the world calls it that.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 21 / 44
27. Proving the Second FTC
b−a
Divide up [a, b] into n pieces of equal width ∆x = as usual. For
n
each i, F is continuous on [xi−1 , xi ] and differentiable on (xi−1 , xi ). So
there is a point ci in (xi−1 , xi ) with
F (xi ) − F (xi−1 )
= F (ci ) = f (ci )
xi − xi−1
Or
f (ci )∆x = F (xi ) − F (xi−1 )
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 22 / 44
28. Proof continued
We have for each i
f (ci )∆x = F (xi ) − F (xi−1 )
Form the Riemann Sum:
n n
Sn = f (ci )∆x = (F (xi ) − F (xi−1 ))
i=1 i=1
= (F (x1 ) − F (x0 )) + (F (x2 ) − F (x1 )) + (F (x3 ) − F (x2 )) + · · ·
· · · + (F (xn−1 ) − F (xn−2 )) + (F (xn ) − F (xn−1 ))
= F (xn ) − F (x0 ) = F (b) − F (a)
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 23 / 44
29. Proof continued
We have for each i
f (ci )∆x = F (xi ) − F (xi−1 )
Form the Riemann Sum:
n n
Sn = f (ci )∆x = (F (xi ) − F (xi−1 ))
i=1 i=1
= (F (x1 ) − F (x0 )) + (F (x2 ) − F (x1 )) + (F (x3 ) − F (x2 )) + · · ·
· · · + (F (xn−1 ) − F (xn−2 )) + (F (xn ) − F (xn−1 ))
= F (xn ) − F (x0 ) = F (b) − F (a)
See if you can spot the invocation of the Mean Value Theorem!
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 23 / 44
30. Proof Completed
We have shown for each n,
Sn = F (b) − F (a)
so in the limit
b
f (x) dx = lim Sn = lim (F (b) − F (a)) = F (b) − F (a)
a n→∞ n→∞
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 24 / 44
31. Computing area with the Second FTC
Example
Find the area between y = x 3 and the x-axis, between x = 0 and x = 1.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 25 / 44
32. Computing area with the Second FTC
Example
Find the area between y = x 3 and the x-axis, between x = 0 and x = 1.
Solution
1 1
x4 1
A= x 3 dx = =
0 4 0 4
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 25 / 44
33. Computing area with the Second FTC
Example
Find the area between y = x 3 and the x-axis, between x = 0 and x = 1.
Solution
1 1
x4 1
A= x 3 dx = =
0 4 0 4
Here we use the notation F (x)|b or [F (x)]b to mean F (b) − F (a).
a a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 25 / 44
34. Computing area with the Second FTC
Example
Find the area enclosed by the parabola y = x 2 and y = 1.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 26 / 44
35. Computing area with the Second FTC
Example
Find the area enclosed by the parabola y = x 2 and y = 1.
1
−1 1
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 26 / 44
36. Computing area with the Second FTC
Example
Find the area enclosed by the parabola y = x 2 and y = 1.
1
−1 1
Solution
1 1
x3 1 1 4
A=2− x 2 dx = 2 − =2− − − =
−1 3 −1 3 3 3
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 26 / 44
37. Computing an integral we estimated before
Example
1
4
Evaluate the integral dx.
0 1 + x2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 27 / 44
39. Computing an integral we estimated before
Example
1
4
Evaluate the integral dx.
0 1 + x2
Solution
1 1
4 1
dx = 4 dx
0 1 + x2 0 1 + x2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 29 / 44
40. Computing an integral we estimated before
Example
1
4
Evaluate the integral dx.
0 1 + x2
Solution
1 1
4 1
dx = 4 dx
0 1 + x2 0 1 + x2
= 4 arctan(x)|1
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 29 / 44
41. Computing an integral we estimated before
Example
1
4
Evaluate the integral dx.
0 1 + x2
Solution
1 1
4 1
dx = 4 dx
0 1 + x2 0 1 + x2
= 4 arctan(x)|1
0
= 4 (arctan 1 − arctan 0)
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 29 / 44
42. Computing an integral we estimated before
Example
1
4
Evaluate the integral dx.
0 1 + x2
Solution
1 1
4 1
dx = 4 dx
0 1 + x2 0 1 + x2
= 4 arctan(x)|1
0
= 4 (arctan 1 − arctan 0)
π
=4 −0
4
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 29 / 44
43. Computing an integral we estimated before
Example
1
4
Evaluate the integral dx.
0 1 + x2
Solution
1 1
4 1
dx = 4 dx
0 1 + x2 0 1 + x2
= 4 arctan(x)|1
0
= 4 (arctan 1 − arctan 0)
π
=4 −0 =π
4
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 29 / 44
44. Computing an integral we estimated before
Example
2
1
Evaluate dx.
1 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 30 / 44
45. Estimating an integral with inequalities
Example
2
1
Estimate dx using Property 8.
1 x
Solution
Since
1 1 1
1 ≤ x ≤ 2 =⇒ ≤ ≤
2 x 1
we have
2
1 1
· (2 − 1) ≤ dx ≤ 1 · (2 − 1)
2 1 x
or
2
1 1
≤ dx ≤ 1
2 1 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 31 / 44
46. Computing an integral we estimated before
Example
2
1
Evaluate dx.
1 x
Solution
2
1
dx
1 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 32 / 44
47. Computing an integral we estimated before
Example
2
1
Evaluate dx.
1 x
Solution
2
1
dx = ln x|2
1
1 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 32 / 44
48. Computing an integral we estimated before
Example
2
1
Evaluate dx.
1 x
Solution
2
1
dx = ln x|2
1
1 x
= ln 2 − ln 1
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 32 / 44
49. Computing an integral we estimated before
Example
2
1
Evaluate dx.
1 x
Solution
2
1
dx = ln x|2
1
1 x
= ln 2 − ln 1
= ln 2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 32 / 44
50. Outline
Last time: The Definite Integral
The definite integral as a limit
Properties of the integral
Comparison Properties of the Integral
Evaluating Definite Integrals
Examples
The Integral as Total Change
Indefinite Integrals
My first table of integrals
Computing Area with integrals
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 33 / 44
51. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 34 / 44
52. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
Theorem
If v (t) represents the velocity of a particle moving rectilinearly, then
t1
v (t) dt = s(t1 ) − s(t0 ).
t0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 34 / 44
53. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
Theorem
If MC (x) represents the marginal cost of making x units of a product, then
x
C (x) = C (0) + MC (q) dq.
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 34 / 44
54. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
Theorem
If ρ(x) represents the density of a thin rod at a distance of x from its end,
then the mass of the rod up to x is
x
m(x) = ρ(s) ds.
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 34 / 44
55. Outline
Last time: The Definite Integral
The definite integral as a limit
Properties of the integral
Comparison Properties of the Integral
Evaluating Definite Integrals
Examples
The Integral as Total Change
Indefinite Integrals
My first table of integrals
Computing Area with integrals
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 35 / 44
56. A new notation for antiderivatives
To emphasize the relationship between antidifferentiation and integration,
we use the indefinite integral notation
f (x) dx
for any function whose derivative is f (x).
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 36 / 44
57. A new notation for antiderivatives
To emphasize the relationship between antidifferentiation and integration,
we use the indefinite integral notation
f (x) dx
for any function whose derivative is f (x). Thus
x 2 dx = 1 x 3 + C .
3
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 36 / 44
58. My first table of integrals
[f (x) + g (x)] dx = f (x) dx + g (x) dx
x n+1
x n dx = + C (n = −1) cf (x) dx = c f (x) dx
n+1
1
e x dx = e x + C dx = ln |x| + C
x
ax
sin x dx = − cos x + C ax dx = +C
ln a
cos x dx = sin x + C csc2 x dx = − cot x + C
sec2 x dx = tan x + C csc x cot x dx = − csc x + C
1
sec x tan x dx = sec x + C √ dx = arcsin x + C
1 − x2
1
dx = arctan x + C
1 + x2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 37 / 44
59. Outline
Last time: The Definite Integral
The definite integral as a limit
Properties of the integral
Comparison Properties of the Integral
Evaluating Definite Integrals
Examples
The Integral as Total Change
Indefinite Integrals
My first table of integrals
Computing Area with integrals
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 38 / 44
60. Example
Find the area between the graph of y = (x − 1)(x − 2), the x-axis, and the
vertical lines x = 0 and x = 3.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 39 / 44
61. Example
Find the area between the graph of y = (x − 1)(x − 2), the x-axis, and the
vertical lines x = 0 and x = 3.
Solution
3
Consider (x − 1)(x − 2) dx.
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 39 / 44
62. Graph
y
x
1 2 3
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 40 / 44
63. Example
Find the area between the graph of y = (x − 1)(x − 2), the x-axis, and the
vertical lines x = 0 and x = 3.
Solution
3
Consider (x − 1)(x − 2) dx. Notice the integrand is positive on [0, 1)
0
and (2, 3], and negative on (1, 2).
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 41 / 44
64. Example
Find the area between the graph of y = (x − 1)(x − 2), the x-axis, and the
vertical lines x = 0 and x = 3.
Solution
3
Consider (x − 1)(x − 2) dx. Notice the integrand is positive on [0, 1)
0
and (2, 3], and negative on (1, 2). If we want the area of the region, we
have to do
1 2 3
A= (x − 1)(x − 2) dx − (x − 1)(x − 2) dx + (x − 1)(x − 2) dx
0 1 2
1 3 1 2 3
= 3x − 2 x 2 + 2x
3
0
− 1 3
3x − 3 x 2 + 2x
2 1
+ 1 3
3x − 2 x 2 + 2x
3
2
5 1 5 11
= − − + = .
6 6 6 6
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 41 / 44
65. Interpretation of “negative area” in motion
There is an analog in rectlinear motion:
t1
v (t) dt is net distance traveled.
t0
t1
|v (t)| dt is total distance traveled.
t0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 42 / 44
66. What about the constant?
It seems we forgot about the +C when we say for instance
1 1
x4 1 1
x 3 dx = = −0=
0 4 0 4 4
But notice
1
x4 1 1 1
+C = +C − (0 + C ) = +C −C =
4 0 4 4 4
no matter what C is.
So in antidifferentiation for definite integrals, the constant is
immaterial.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 43 / 44
67. Summary
The second Fundamental Theorem of Calculus:
b
f (x) dx = F (b) − F (a)
a
where F = f .
Definite integrals represent net change of a function over an interval.
We write antiderivatives as indefinite integrals f (x) dx
V63.0121.002.2010Su, Calculus I (NYU) Section 5.3 Evaluating Definite Integrals June 21, 2010 44 / 44