We trace the computation of area through the centuries. The process known known as Riemann Sums has applications to not just area but many fields of science.
We trace the computation of area through the centuries. The process known known as Riemann Sums has applications to not just area but many fields of science.
(Handout version of slideshow from class)
This document discusses wavefront analysis for an annular ellipse aperture using Zernike polynomials. It begins with an introduction to Zernike polynomials for different apertures developed by other researchers. The relationship between Zernike annular ellipse polynomials and third-order Seidel aberrations is then derived. Finally, the standard deviations of balanced and unbalanced primary aberrations are calculated for an annular ellipse aperture using the property that Zernike polynomials form an orthogonal set over the aperture.
Fundamental Geometrical Concepts Class 7Tushar Gupta
I made this presentation for my school project after that I thought that I should upload it on any slide so I uploaded this to help others in making presentations and getting ideas.It is a class 7 project.
This document defines key geometric concepts including points, lines, planes and their relationships. It explains that points have no size, lines extend indefinitely and planes are flat surfaces that extend without limits. It also covers topics like collinear points that lie on the same line, determining if objects are coplanar by lying on the same plane, and the different types of intersections between lines and planes, which can be a point, line or no intersection.
Some devices could not work corrctly on A.C (Alternating current source) thats why we need to Convert A.C(Alternating current source) into D.C (Direct current source).
Effective two terminal single line to ground fault location algorithmMuhd Hafizi Idris
This paper presents an effective algorithm to locate Single Line to Ground (SLG) fault at a transmission line. Post fault voltages and currents from both substation terminals were used as the input parameters to the algorithm. Discrete Fourier Transform (DFT) was used to extract the magnitudes and phase angles of three phase voltages and currents. The modeling of the transmission line along with the algorithm was performed using Matlab/Simulink package. The results of fault location for SLG faults along the transmission line demonstrated the validity of the algorithm used even for high resistance earth fault.
The document introduces some basic concepts in geometry, including:
1. Points, lines, and planes are undefined terms that form the foundations of geometry.
2. It explains concepts like collinear points, coplanar points, line segments, rays, and how to classify angles.
3. It discusses intersections of lines, planes, and examples of modeling intersections of geometric figures.
Communication Engineering - Chapter 6 - Noisemkazree
This document discusses various types of noise that can interfere with communication signals. It defines noise and categorizes it as either correlated noise, which depends on the presence of a signal, or uncorrelated noise, which is always present. Examples of uncorrelated noise sources include atmospheric noise from lightning, extraterrestrial noise from the sun and stars, industrial noise from machinery, thermal noise from component movement, and shot noise from random carrier arrival. The effects of noise on signals and ways to measure noise are also summarized.
We trace the computation of area through the centuries. The process known known as Riemann Sums has applications to not just area but many fields of science.
(Handout version of slideshow from class)
This document discusses wavefront analysis for an annular ellipse aperture using Zernike polynomials. It begins with an introduction to Zernike polynomials for different apertures developed by other researchers. The relationship between Zernike annular ellipse polynomials and third-order Seidel aberrations is then derived. Finally, the standard deviations of balanced and unbalanced primary aberrations are calculated for an annular ellipse aperture using the property that Zernike polynomials form an orthogonal set over the aperture.
Fundamental Geometrical Concepts Class 7Tushar Gupta
I made this presentation for my school project after that I thought that I should upload it on any slide so I uploaded this to help others in making presentations and getting ideas.It is a class 7 project.
This document defines key geometric concepts including points, lines, planes and their relationships. It explains that points have no size, lines extend indefinitely and planes are flat surfaces that extend without limits. It also covers topics like collinear points that lie on the same line, determining if objects are coplanar by lying on the same plane, and the different types of intersections between lines and planes, which can be a point, line or no intersection.
Some devices could not work corrctly on A.C (Alternating current source) thats why we need to Convert A.C(Alternating current source) into D.C (Direct current source).
Effective two terminal single line to ground fault location algorithmMuhd Hafizi Idris
This paper presents an effective algorithm to locate Single Line to Ground (SLG) fault at a transmission line. Post fault voltages and currents from both substation terminals were used as the input parameters to the algorithm. Discrete Fourier Transform (DFT) was used to extract the magnitudes and phase angles of three phase voltages and currents. The modeling of the transmission line along with the algorithm was performed using Matlab/Simulink package. The results of fault location for SLG faults along the transmission line demonstrated the validity of the algorithm used even for high resistance earth fault.
The document introduces some basic concepts in geometry, including:
1. Points, lines, and planes are undefined terms that form the foundations of geometry.
2. It explains concepts like collinear points, coplanar points, line segments, rays, and how to classify angles.
3. It discusses intersections of lines, planes, and examples of modeling intersections of geometric figures.
Communication Engineering - Chapter 6 - Noisemkazree
This document discusses various types of noise that can interfere with communication signals. It defines noise and categorizes it as either correlated noise, which depends on the presence of a signal, or uncorrelated noise, which is always present. Examples of uncorrelated noise sources include atmospheric noise from lightning, extraterrestrial noise from the sun and stars, industrial noise from machinery, thermal noise from component movement, and shot noise from random carrier arrival. The effects of noise on signals and ways to measure noise are also summarized.
The document provides an overview of measurement system analysis (MSA) techniques for both variable and attribute gages. It describes the average-range method and ANOVA method for analyzing variable gages, and the short method, hypothesis test analysis, and long method for attribute gages. Acceptability criteria are outlined for determining if a measurement system is capable of measuring process variation.
Vectors have both magnitude and direction, represented by arrows. The sum of two vectors is obtained by placing the tail of one vector at the head of the other. If the vectors are at right angles, their dot product is zero, while their cross product is maximum. Scalar multiplication scales the magnitude but not the direction of a vector.
The document discusses different types of noise that affect communication systems, including thermal noise, shot noise, flicker noise, excess resistor noise, and popcorn noise. It provides details on thermal noise generation and its relation to temperature and resistance. The analysis section examines thermal noise in resistors in series and parallel and defines signal-to-noise ratio and noise factor. Additive white Gaussian noise is described as noise that is additive, has a constant spectral density (white), and has a Gaussian amplitude distribution.
1) Noise exists in all communication systems and degrades signal quality. It is caused by random movement of electrons and can be internal or external.
2) Thermal noise, also known as Johnson noise, is generated by thermal agitation of electrons in conductors. It is proportional to temperature and bandwidth.
3) Noise figure and noise temperature are used to measure the degradation of signal to noise ratio caused by components in a communication system. Lower noise figure and temperature indicate less degradation.
The document discusses basic concepts in geometry including points, lines, planes, and their relationships. It defines a point as having no size or shape, a line as connecting two or more points and extending indefinitely in both directions, and a plane as a flat two-dimensional surface containing points and lines. The document provides examples of naming points, lines, and planes and identifies collinear points that lie on the same line and coplanar points that lie on the same plane. It includes practice problems asking students to name, draw, and identify various geometric concepts.
Geometry is the branch of mathematics concerned with properties of points, lines, angles, curves, surfaces and solids. It involves visualizing shapes, sizes, patterns and positions. The presentation introduced basic concepts like different types of lines, rays and angles. It also discussed plane figures from kindergarten to 8th grade, including classifying shapes by number of sides. Space figures like cubes and pyramids were demonstrated by having students construct 3D models. The concepts of tessellation, symmetry, and line of symmetry were explained.
The document discusses strategies for teaching mathematics, including discovery approach, inquiry teaching, demonstration approach, math-lab approach, practical work approach, individualized instruction using modules, brainstorming, problem-solving, cooperative learning, and integrative technique. It provides details on each approach, such as the discovery approach aiming to develop higher-order thinking skills and both teachers and learners playing active roles. It also lists 10 creative ways to teach math using dramatizations, children's bodies, play, toys, stories, creativity, and problem-solving abilities.
Lesson 24: Areas, Distances, the Integral (Section 041 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.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
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.
Lesson 24: Areas, Distances, the Integral (Section 041 handout)Matthew Leingang
This document outlines the key concepts and objectives from Sections 5.1-5.2 of a Calculus I course. It discusses how areas and distances can be computed using Riemann sums and the definite integral. Specifically, it covers how areas of curved regions were approximated historically from Euclid to Cavalieri using inscribed polygons. It also discusses how the definite integral can be used to compute distances traveled using the rate-time analogy to Riemann sums. The goal is to introduce computing areas and distances as limits of Riemann sums.
This document provides information about traversing and site surveying techniques. It discusses open, closed, and connecting traverses. It outlines the components and functions of surveying equipment used, including theodolites, tripods, level rods, and plummets. The document describes procedures for traversing fieldwork such as collecting angle and distance measurements, calculating bearings, latitudes and departures, and adjusting for closure errors. It also provides objectives and guidelines for station selection and traverse calculations to determine coordinate positions and ensure survey accuracy.
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 summarizes the problems from the IV International Olympiad in Physics held in Moscow, USSR in 1970. Teams from 8 countries participated in theoretical and experimental competitions. The theoretical problems covered mechanics, crystal structure, electrostatics, and optics. The experimental problem involved determining focal lengths of lenses. The solutions provided the calculations and reasoning to solve each theoretical problem in under 3 sentences.
This document describes the analysis of high-quality X-ray spectra of Mrk 509 taken with the Reflection Grating Spectrometers on XMM-Newton. The spectra were obtained using a new multi-pointing mode over 600 ks to constrain properties of the outflow. Combining the individual spectra required developing new methods to account for gaps in the data from detector issues. Absorption lines in the spectra were analyzed to study the outflow.
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.
This document provides details on a field work report for a traverse survey conducted in August 2015. It includes an introduction to traversing and different types of traverses. It outlines the equipment used including a theodolite, tripod, ranging poles, tapes, and other accessories. The document describes the objectives and field data collection process. It provides steps for computations including balancing angles, determining line directions, calculating latitudes and departures, adjusting misclosures, and determining coordinates. A conclusion discusses the results of the traverse survey.
Imaging the dust_sublimation_front_of_a_circumbinary_diskSérgio Sacani
Aims. We present the first near-IR milli-arcsecond-scale image of a post-AGB binary that is surrounded by hot circumbinary dust.
Methods. A very rich interferometric data set in six spectral channels was acquired of IRAS 08544-4431 with the new RAPID camera
on the PIONIER beam combiner at the Very Large Telescope Interferometer (VLTI). A broadband image in the H-band was reconstructed
by combining the data of all spectral channels using the SPARCO method.
Results. We spatially separate all the building blocks of the IRAS 08544-4431 system in our milliarcsecond-resolution image. Our
dissection reveals a dust sublimation front that is strikingly similar to that expected in early-stage protoplanetary disks, as well as an
unexpected flux signal of 4% from the secondary star. The energy output from this companion indicates the presence of a compact
circum-companion accretion disk, which is likely the origin of the fast outflow detected in H.
Conclusions. Our image provides the most detailed view into the heart of a dusty circumstellar disk to date. Our results demonstrate
that binary evolution processes and circumstellar disk evolution can be studied in detail in space and over time.
The document discusses crystal indexing and diffraction. It introduces Miller indices for labeling crystal planes, how to calculate d-spacings, and Bragg's law for deriving x-ray diffraction from crystal planes. The objectives are to understand plane identification in crystals, calculate Miller indices and d-spacings, and derive and use Bragg's law to solve diffraction problems. Examples are provided to illustrate key concepts and calculations.
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.
The document provides an overview of measurement system analysis (MSA) techniques for both variable and attribute gages. It describes the average-range method and ANOVA method for analyzing variable gages, and the short method, hypothesis test analysis, and long method for attribute gages. Acceptability criteria are outlined for determining if a measurement system is capable of measuring process variation.
Vectors have both magnitude and direction, represented by arrows. The sum of two vectors is obtained by placing the tail of one vector at the head of the other. If the vectors are at right angles, their dot product is zero, while their cross product is maximum. Scalar multiplication scales the magnitude but not the direction of a vector.
The document discusses different types of noise that affect communication systems, including thermal noise, shot noise, flicker noise, excess resistor noise, and popcorn noise. It provides details on thermal noise generation and its relation to temperature and resistance. The analysis section examines thermal noise in resistors in series and parallel and defines signal-to-noise ratio and noise factor. Additive white Gaussian noise is described as noise that is additive, has a constant spectral density (white), and has a Gaussian amplitude distribution.
1) Noise exists in all communication systems and degrades signal quality. It is caused by random movement of electrons and can be internal or external.
2) Thermal noise, also known as Johnson noise, is generated by thermal agitation of electrons in conductors. It is proportional to temperature and bandwidth.
3) Noise figure and noise temperature are used to measure the degradation of signal to noise ratio caused by components in a communication system. Lower noise figure and temperature indicate less degradation.
The document discusses basic concepts in geometry including points, lines, planes, and their relationships. It defines a point as having no size or shape, a line as connecting two or more points and extending indefinitely in both directions, and a plane as a flat two-dimensional surface containing points and lines. The document provides examples of naming points, lines, and planes and identifies collinear points that lie on the same line and coplanar points that lie on the same plane. It includes practice problems asking students to name, draw, and identify various geometric concepts.
Geometry is the branch of mathematics concerned with properties of points, lines, angles, curves, surfaces and solids. It involves visualizing shapes, sizes, patterns and positions. The presentation introduced basic concepts like different types of lines, rays and angles. It also discussed plane figures from kindergarten to 8th grade, including classifying shapes by number of sides. Space figures like cubes and pyramids were demonstrated by having students construct 3D models. The concepts of tessellation, symmetry, and line of symmetry were explained.
The document discusses strategies for teaching mathematics, including discovery approach, inquiry teaching, demonstration approach, math-lab approach, practical work approach, individualized instruction using modules, brainstorming, problem-solving, cooperative learning, and integrative technique. It provides details on each approach, such as the discovery approach aiming to develop higher-order thinking skills and both teachers and learners playing active roles. It also lists 10 creative ways to teach math using dramatizations, children's bodies, play, toys, stories, creativity, and problem-solving abilities.
Lesson 24: Areas, Distances, the Integral (Section 041 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.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
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.
Lesson 24: Areas, Distances, the Integral (Section 041 handout)Matthew Leingang
This document outlines the key concepts and objectives from Sections 5.1-5.2 of a Calculus I course. It discusses how areas and distances can be computed using Riemann sums and the definite integral. Specifically, it covers how areas of curved regions were approximated historically from Euclid to Cavalieri using inscribed polygons. It also discusses how the definite integral can be used to compute distances traveled using the rate-time analogy to Riemann sums. The goal is to introduce computing areas and distances as limits of Riemann sums.
This document provides information about traversing and site surveying techniques. It discusses open, closed, and connecting traverses. It outlines the components and functions of surveying equipment used, including theodolites, tripods, level rods, and plummets. The document describes procedures for traversing fieldwork such as collecting angle and distance measurements, calculating bearings, latitudes and departures, and adjusting for closure errors. It also provides objectives and guidelines for station selection and traverse calculations to determine coordinate positions and ensure survey accuracy.
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 summarizes the problems from the IV International Olympiad in Physics held in Moscow, USSR in 1970. Teams from 8 countries participated in theoretical and experimental competitions. The theoretical problems covered mechanics, crystal structure, electrostatics, and optics. The experimental problem involved determining focal lengths of lenses. The solutions provided the calculations and reasoning to solve each theoretical problem in under 3 sentences.
This document describes the analysis of high-quality X-ray spectra of Mrk 509 taken with the Reflection Grating Spectrometers on XMM-Newton. The spectra were obtained using a new multi-pointing mode over 600 ks to constrain properties of the outflow. Combining the individual spectra required developing new methods to account for gaps in the data from detector issues. Absorption lines in the spectra were analyzed to study the outflow.
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.
This document provides details on a field work report for a traverse survey conducted in August 2015. It includes an introduction to traversing and different types of traverses. It outlines the equipment used including a theodolite, tripod, ranging poles, tapes, and other accessories. The document describes the objectives and field data collection process. It provides steps for computations including balancing angles, determining line directions, calculating latitudes and departures, adjusting misclosures, and determining coordinates. A conclusion discusses the results of the traverse survey.
Imaging the dust_sublimation_front_of_a_circumbinary_diskSérgio Sacani
Aims. We present the first near-IR milli-arcsecond-scale image of a post-AGB binary that is surrounded by hot circumbinary dust.
Methods. A very rich interferometric data set in six spectral channels was acquired of IRAS 08544-4431 with the new RAPID camera
on the PIONIER beam combiner at the Very Large Telescope Interferometer (VLTI). A broadband image in the H-band was reconstructed
by combining the data of all spectral channels using the SPARCO method.
Results. We spatially separate all the building blocks of the IRAS 08544-4431 system in our milliarcsecond-resolution image. Our
dissection reveals a dust sublimation front that is strikingly similar to that expected in early-stage protoplanetary disks, as well as an
unexpected flux signal of 4% from the secondary star. The energy output from this companion indicates the presence of a compact
circum-companion accretion disk, which is likely the origin of the fast outflow detected in H.
Conclusions. Our image provides the most detailed view into the heart of a dusty circumstellar disk to date. Our results demonstrate
that binary evolution processes and circumstellar disk evolution can be studied in detail in space and over time.
The document discusses crystal indexing and diffraction. It introduces Miller indices for labeling crystal planes, how to calculate d-spacings, and Bragg's law for deriving x-ray diffraction from crystal planes. The objectives are to understand plane identification in crystals, calculate Miller indices and d-spacings, and derive and use Bragg's law to solve diffraction problems. Examples are provided to illustrate key concepts and calculations.
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.
Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 22: Areas and Distances
1. Section 5.1
Areas and Distances
V63.0121.006/016, Calculus I
New York University
April 13, 2010
Announcements
Quiz April 16 on §§4.1–4.4
Final Exam: Monday, May 10, 12:00noon
2. Announcements
Quiz April 16 on §§4.1–4.4
Final Exam: Monday, May
10, 12:00noon
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 2 / 30
3. Objectives
Compute the area of a
region by approximating it
with rectangles and letting
the size of the rectangles
tend to zero.
Compute the total distance
traveled by a particle by
approximating it as distance
= (rate)(time) and letting
the time intervals over which
one approximates tend to
zero.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 3 / 30
4. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 4 / 30
5. Easy Areas: Rectangle
Definition
The area of a rectangle with dimensions and w is the product A = w .
w
It may seem strange that this is a definition and not a theorem but we
have to start somewhere.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 5 / 30
6. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
b
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 6 / 30
7. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
b
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 6 / 30
8. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 6 / 30
9. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
b
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 6 / 30
10. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
b
So
Fact
The area of a parallelogram of base width b and height h is
A = bh
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 6 / 30
11. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a parallelogram.
b
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 7 / 30
12. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a parallelogram.
h
b
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 7 / 30
13. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a parallelogram.
h
b
So
Fact
The area of a triangle of base width b and height h is
1
A = bh
2
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 7 / 30
14. Easy Areas: Other Polygons
Any polygon can be triangulated, so its area can be found by summing the
areas of the triangles:
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 8 / 30
15. Hard Areas: Curved Regions
???
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 9 / 30
16. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC –
212 BC (after Euclid)
Geometer
Weapons engineer
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 10 / 30
17. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC –
212 BC (after Euclid)
Geometer
Weapons engineer
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 10 / 30
18. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC –
212 BC (after Euclid)
Geometer
Weapons engineer
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 10 / 30
19. Archimedes found areas of a sequence of triangles inscribed in a parabola.
A=
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 11 / 30
20. 1
Archimedes found areas of a sequence of triangles inscribed in a parabola.
A=1
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 11 / 30
21. 1
1 1
8 8
Archimedes found areas of a sequence of triangles inscribed in a parabola.
1
A=1+2·
8
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 11 / 30
22. 1 1
64 64
1
1 1
8 8
1 1
64 64
Archimedes found areas of a sequence of triangles inscribed in a parabola.
1 1
A=1+2· +4· + ···
8 64
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 11 / 30
23. 1 1
64 64
1
1 1
8 8
1 1
64 64
Archimedes found areas of a sequence of triangles inscribed in a parabola.
1 1
A=1+2· +4· + ···
8 64
1 1 1
=1+ + + ··· + n + ···
4 16 4
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 11 / 30
24. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 12 / 30
25. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
But for any number r and any positive integer n,
(1 − r )(1 + r + · · · + r n ) = 1 − r n+1
So
1 − r n+1
1 + r + · · · + rn =
1−r
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 12 / 30
26. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
But for any number r and any positive integer n,
(1 − r )(1 + r + · · · + r n ) = 1 − r n+1
So
1 − r n+1
1 + r + · · · + rn =
1−r
Therefore
1 1 1 1 − (1/4)n+1
1+ + + ··· + n =
4 16 4 1 − 1/4
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 12 / 30
27. We would then need to know the value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
But for any number r and any positive integer n,
(1 − r )(1 + r + · · · + r n ) = 1 − r n+1
So
1 − r n+1
1 + r + · · · + rn =
1−r
Therefore
1 1 1 1 − (1/4)n+1 1 4
1+ + + ··· + n = →3 =
4 16 4 1 − 1/4 /4 3
as n → ∞.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 12 / 30
28. Cavalieri
Italian,
1598–1647
Revisited the
area problem
with a
different
perspective
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 13 / 30
29. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
0 1
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
30. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
0 1 1
2
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
31. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
L3 =
0 1 2 1
3 3
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
32. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
0 1 2 1
3 3
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
33. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
L4 =
0 1 2 3 1
4 4 4
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
34. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
64 64 64 64
0 1 2 3 1
4 4 4
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
35. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
64 64 64 64
0 1 2 3 4 1 L5 =
5 5 5 5
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
36. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
64 64 64 64
1 4 9 16 30
0 1 2 3 4 1 L5 = + + + =
125 125 125 125 125
5 5 5 5
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
37. Cavalieri’s method
Divide up the interval into pieces
y = x2 and measure the area of the
inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
1 4 9 14
L4 = + + =
64 64 64 64
1 4 9 16 30
0 1 L5 = + + + =
125 125 125 125 125
Ln =?
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 14 / 30
38. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 15 / 30
39. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
2
1 i −1 (i − 1)2
· = .
n n n3
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 15 / 30
40. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
2
1 i −1 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 15 / 30
41. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
2
1 i −1 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
The Arabs knew that
n(n − 1)(2n − 1)
1 + 22 + 32 + · · · + (n − 1)2 =
6
So
n(n − 1)(2n − 1)
Ln =
6n3
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 15 / 30
42. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width . The
n
rectangle over the ith interval and under the parabola has area
2
1 i −1 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
The Arabs knew that
n(n − 1)(2n − 1)
1 + 22 + 32 + · · · + (n − 1)2 =
6
So
n(n − 1)(2n − 1) 1
Ln = →
6n3 3
as n → ∞.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 15 / 30
43. Cavalieri’s method for different functions
Try the same trick with f (x) = x 3 . We have
1 1 1 2 1 n−1
Ln = ·f + ·f + ··· + ·f
n n n n n n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 16 / 30
44. Cavalieri’s method for different functions
Try the same trick with f (x) = x 3 . We have
1 1 1 2 1 n−1
Ln = ·f + ·f + ··· + · f
n n n n n n
1 1 1 2 3 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 16 / 30
45. Cavalieri’s method for different functions
Try the same trick with f (x) = x 3 . We have
1 1 1 2 1 n−1
Ln = ·f + ·f + ··· + · f
n n n n n n
1 1 1 2 3 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
1+2 3 + 33 + · · · + (n − 1)3
=
n4
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 16 / 30
46. Cavalieri’s method for different functions
Try the same trick with f (x) = x 3 . We have
1 1 1 2 1 n−1
Ln = ·f + ·f + ··· + · f
n n n n n n
1 1 1 2 3 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
1+2 3 + 33 + · · · + (n − 1)3
=
n4
The formula out of the hat is
2
1 + 23 + 33 + · · · + (n − 1)3 = 1
2 n(n − 1)
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 16 / 30
47. Cavalieri’s method for different functions
Try the same trick with f (x) = x 3 . We have
1 1 1 2 1 n−1
Ln = ·f + ·f + ··· + · f
n n n n n n
1 1 1 2 3 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
1+2 3 + 33 + · · · + (n − 1)3
=
n4
The formula out of the hat is
2
1 + 23 + 33 + · · · + (n − 1)3 = 1
2 n(n − 1)
So
n2 (n − 1)2 1
Ln = →
4n4 4
as n → ∞.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 16 / 30
48. Cavalieri’s method with different heights
1 13 1 23 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
13 + 23 + 33 + · · · + n3
=
n4
1 1 2
= 4 2 n(n + 1)
n
n2 (n + 1)2 1
= →
4n4 4
as n → ∞.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 17 / 30
49. Cavalieri’s method with different heights
1 13 1 23 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
13 + 23 + 33 + · · · + n3
=
n4
1 1 2
= 4 2 n(n + 1)
n
n2 (n + 1)2 1
= →
4n4 4
as n → ∞.
So even though the rectangles overlap, we still get the same answer.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 17 / 30
50. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 18 / 30
51. Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want to find the area
between x = a, x = b, y = 0, and y = f (x).
b−a
For each positive integer n, divide up the interval into n pieces. Then ∆x = .
n
For each i between 1 and n, let xi be the nth step between a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 ·
n
······
b−a
xi = a + i ·
n
······
a b b−a
x0 x1 x2 . . . xi xn−1xn xn = a + n · =b
n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 19 / 30
52. Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want to find the area
between x = a, x = b, y = 0, and y = f (x).
b−a
For each positive integer n, divide up the interval into n pieces. Then ∆x = .
n
For each i between 1 and n, let xi be the nth step between a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 ·
n
······
b−a
xi = a + i ·
n
······
a b b−a
x0 x1 x2 . . . xi xn−1xn xn = a + n · =b
n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 19 / 30
53. Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want to find the area
between x = a, x = b, y = 0, and y = f (x).
b−a
For each positive integer n, divide up the interval into n pieces. Then ∆x = .
n
For each i between 1 and n, let xi be the nth step between a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 ·
n
······
b−a
xi = a + i ·
n
······
a b b−a
x0 x1 x2 . . . xi xn−1xn xn = a + n · =b
n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 19 / 30
54. Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want to find the area
between x = a, x = b, y = 0, and y = f (x).
b−a
For each positive integer n, divide up the interval into n pieces. Then ∆x = .
n
For each i between 1 and n, let xi be the nth step between a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 ·
n
······
b−a
xi = a + i ·
n
······
a b b−a
x0 x1 x2 . . . xi xn−1xn xn = a + n · =b
n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 19 / 30
55. Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want to find the area
between x = a, x = b, y = 0, and y = f (x).
b−a
For each positive integer n, divide up the interval into n pieces. Then ∆x = .
n
For each i between 1 and n, let xi be the nth step between a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 ·
n
······
b−a
xi = a + i ·
n
······
a b b−a
x0 x1 x2 . . . xi xn−1xn xn = a + n · =b
n
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 19 / 30
56. Forming Riemann sums
We have many choices of how to approximate the area:
Ln = f (x0 )∆x + f (x1 )∆x + · · · + f (xn−1 )∆x
Rn = f (x1 )∆x + f (x2 )∆x + · · · + f (xn )∆x
x0 + x1 x1 + x2 xn−1 + xn
Mn = f ∆x + f ∆x + · · · + f ∆x
2 2 2
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 20 / 30
57. Forming Riemann sums
We have many choices of how to approximate the area:
Ln = f (x0 )∆x + f (x1 )∆x + · · · + f (xn−1 )∆x
Rn = f (x1 )∆x + f (x2 )∆x + · · · + f (xn )∆x
x0 + x1 x1 + x2 xn−1 + xn
Mn = f ∆x + f ∆x + · · · + f ∆x
2 2 2
In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the
Riemann sum
Sn = f (c1 )∆x + f (c2 )∆x + · · · + f (cn )∆x
n
= f (ci )∆x
i=1
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 20 / 30
58. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a b
x1
matter what choice of ci we
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
59. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x1 b
x2
matter what choice of ci we
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
60. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x1 x2 b
x3
matter what choice of ci we
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
61. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x1 x2 x3 b
x4
matter what choice of ci we
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
62. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x x x x x
matter what choice of ci we 1 2 3 4 b5
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
63. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x x x x x x
matter what choice of ci we 1 2 3 4 5 b6
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
64. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x x x x x x x
matter what choice of ci we 1 2 3 4 5 6 b7
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
65. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x
matter what choice of ci we 1 2 3 4 5 6 7 b8
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
66. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x x
matter what choice of ci we 1 2 3 4 5 6 7 8b9
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
67. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a x x x x x x x x x xb
matter what choice of ci we 1 2 3 4 5 6 7 8 9 10
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
68. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x x x xb
matter what choice of ci we 1 2 3 4 5 6 7 8 9 1011
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
69. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x xx x xb
matter what choice of ci we 1 2 3 4 5 6 7 8 9 101112
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
70. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x xx x x xb
matter what choice of ci we 1 2 3 4 5 6 7 8 910 12
11 13
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
71. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x xx x x x xb
matter what choice of ci we 1 2 3 4 5 6 7 8 910 12 14
11 13
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
72. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
ax x x x x x x x xx x x x x xb
matter what choice of ci we 1 2 3 4 5 6 7 8 910 12 14
11 13 15
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
73. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
axxxxxxxxxxxxxxxxb
matter what choice of ci we 1 2 3 4 5 6 7 8 910 12 14 16
11 13 15
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
74. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a xxxxxxxxxxxxxxxxb
x1 2 3 4 5 6 7 8 910 12 14 16
matter what choice of ci we 11 13 15 17
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
75. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a xxxxxxxx xxxxxxxxb
x1 2 3 4 5 6 7 8x10 12 14 16 18
matter what choice of ci we 9 11 13 15 17
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
76. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a xxxxxxxx xxxxxxxxxb
x12345678910 12 14 16 18
x 11 13 15 17 19
matter what choice of ci we
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
77. Theorem of the Day
Theorem
If f is a continuous function or
has finitely many jump
discontinuities on [a, b], then
n
lim Sn = lim f (ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
a xxxxxxxx xxxxxxxxxxb
x123456789101214161820
x 1113151719
matter what choice of ci we
made.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 21 / 30
78. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4)
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
79. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4)
Want the slope of a curve
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
80. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4) Want the area of a curved
region
Want the slope of a curve
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
81. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4) Want the area of a curved
region
Want the slope of a curve
Only know the slope of lines
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
82. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4) Want the area of a curved
region
Want the slope of a curve
Only know the area of
Only know the slope of lines
polygons
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
83. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4) Want the area of a curved
region
Want the slope of a curve
Only know the area of
Only know the slope of lines
polygons
Approximate curve with a
line
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
84. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4) Want the area of a curved
region
Want the slope of a curve
Only know the area of
Only know the slope of lines
polygons
Approximate curve with a
Approximate region with
line
polygons
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
85. Analogies
The Area Problem (Ch. 5)
The Tangent Problem
(Ch. 2–4) Want the area of a curved
region
Want the slope of a curve
Only know the area of
Only know the slope of lines
polygons
Approximate curve with a
Approximate region with
line
polygons
Take limit over better and
Take limit over better and
better approximations
better approximations
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 22 / 30
86. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 23 / 30
87. Distances
Just like area = length × width, we have
distance = rate × time.
So here is another use for Riemann sums.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 24 / 30
89. Example
A sailing ship is cruising back and forth along a channel (in a straight
line). At noon the ship’s position and velocity are recorded, but shortly
thereafter a storm blows in and position is impossible to measure. The
velocity continues to be recorded at thirty-minute intervals.
Time 12:00 12:30 1:00 1:30 2:00
Speed (knots) 4 8 12 6 4
Direction E E E E W
Time 2:30 3:00 3:30 4:00
Speed 3 3 5 9
Direction W E E E
Estimate the ship’s position at 4:00pm.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 26 / 30
90. Solution
We estimate that the speed of 4 knots (nautical miles per hour) is
maintained from 12:00 until 12:30. So over this time interval the ship
travels
4 nmi 1
hr = 2 nmi
hr 2
We can continue for each additional half hour and get
distance = 4 × 1/2 + 8 × 1/2 + 12 × 1/2
+ 6 × 1/2 − 4 × 1/2 − 3 × 1/2 + 3 × 1/2 + 5 × 1/2
= 15.5
So the ship is 15.5 nmi east of its original position.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 27 / 30
91. Analysis
This method of measuring position by recording velocity was necessary
until global-positioning satellite technology became widespread
If we had velocity estimates at finer intervals, we’d get better
estimates.
If we had velocity at every instant, a limit would tell us our exact
position relative to the last time we measured it.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 28 / 30
92. Other uses of Riemann sums
Anything with a product!
Area, volume
Anything with a density: Population, mass
Anything with a “speed:” distance, throughput, power
Consumer surplus
Expected value of a random variable
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 29 / 30
93. Summary
We can compute the area of a curved region with a limit of Riemann
sums
We can compute the distance traveled from the velocity with a limit
of Riemann sums
Many other important uses of this process.
V63.0121.006/016, Calculus I (NYU) Section 5.1 Areas and Distances April 13, 2010 30 / 30