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.
Structural measurements in oriented core photograph january 2019_galkineVadim Galkine
n this post I describe the method of structural measurements of planar structures using oriented core digital photographs. The main advantage of the method is an opportunity to reduce field-based time of the drill-core processing. All the measurements can be done in the office.
Users can work either in the standard GIS platforms (ArcMap, MapInfo etc) or even use digitizers outside of GIS environment which makes the technique comparatively cheap.
The method consists of two steps:
1) digitazing photographs and obtaining a table of xy coordinates of the three-point sets of planar structures
2) calculating actual structure orientations using the MSExcel calculation spreadsheets.
The spreadsheets are provided as attachments to the post. They can also be downloaded from http://remoteexploration.com/oriented-core-techniques.html
Dynamic texture based traffic vehicle monitoring systemeSAT Journals
Abstract Dynamic Textures are function of both space and time which exhibit spatio-temporal stationary property. They have spatially repetitive patterns which vary with time. In this paper, importance of phase spectrum in the signals is utilized and a novel method for vehicle monitoring is proposed with the help of Fourier analysis, synthesis and image processing. Methods like Doppler radar or GPS navigation are used commonly for tracking. The proposed image based approach has an added advantage that the clear image of the object (vehicle) can be used for future reference like proof of incidence, identification of owner and registration number. Keywords-Fourier Transform, dynamic texture, phase spectrum
Localization of free 3 d surfaces by the mean of photometricIAEME Publication
This document discusses a photometric stereovision technique to localize free 3D surfaces using multiple images.
It presents a photometric model linking image intensity to surface relief and reflectance properties. This model results in a system of equations with the relief variations and reflectance as unknowns. Solving the system requires 3 images under different lighting conditions.
The document applies this technique to extract the 3D relief of a corrugated plastic surface from images, demonstrating the feasibility of measuring free surfaces with the photometric stereovision approach. Accuracy of the measured surface shape is evaluated.
This document summarizes the principles of photogrammetry. It discusses the basic elements of photogrammetry including obtaining quantitative information from aerial photographs. It covers topics such as photographic scale, horizontal ground coordinates, relief displacement, exterior orientation of tilted photographs, stereoscopic vision, and the geometry of aerial stereophotographs. The purpose is to provide background information and references to support standards and guidelines for photogrammetric mapping.
Photogrammetry is the science of obtaining information about physical objects through photographs, without needing direct contact. It involves measuring and analyzing captured images. The name comes from Greek roots meaning "light", "drawing", and "to measure". Key developments included using photography for mapmaking in the 1840s-1850s, the photogrammetric stereoplotter in the 1890s, and aerial photography from balloons and planes in the 1860s-1900s, advancing the field into the digital era.
The document summarizes key concepts about aerotriangulation from a lecture on photogrammetry. It discusses that aerotriangulation establishes geometric relationships between overlapping photographs to determine supplemental ground control points. Aerotriangulation can be performed using semianalytical or analytical bundle block methods. The analytical bundle block method represents the geometric relationships between object space, camera positions, and image points in a mathematical model.
Photogrammetry is a technique for obtaining reliable spatial information about physical objects through analyzing photographs. It involves taking overlapping aerial photographs from aircraft and processing them using software to extract 3D spatial data like digital elevation models, contours, and orthophotos. The key aspects are that it is precise, cost-effective, produces 3D representations, and relies on established algorithms to extract data from overlapping photos with less manual effort than traditional surveying methods.
Method of Fracture Surface Matching Based on Mathematical StatisticsIJRESJOURNAL
ABSTRACT: Fracture surface matching is an important part of point cloud registration. In this paper, a method of fracture surface matching based on mathematical statistics is proposed. We reconstruct a coordinate system of the fractured surface points, and analyze the characteristics of the point cloud in the new coordinate system, using the theory of mathematical statistcs. The general distribution of the points is determined. The method can realize the matching relation among some point cloud.
Structural measurements in oriented core photograph january 2019_galkineVadim Galkine
n this post I describe the method of structural measurements of planar structures using oriented core digital photographs. The main advantage of the method is an opportunity to reduce field-based time of the drill-core processing. All the measurements can be done in the office.
Users can work either in the standard GIS platforms (ArcMap, MapInfo etc) or even use digitizers outside of GIS environment which makes the technique comparatively cheap.
The method consists of two steps:
1) digitazing photographs and obtaining a table of xy coordinates of the three-point sets of planar structures
2) calculating actual structure orientations using the MSExcel calculation spreadsheets.
The spreadsheets are provided as attachments to the post. They can also be downloaded from http://remoteexploration.com/oriented-core-techniques.html
Dynamic texture based traffic vehicle monitoring systemeSAT Journals
Abstract Dynamic Textures are function of both space and time which exhibit spatio-temporal stationary property. They have spatially repetitive patterns which vary with time. In this paper, importance of phase spectrum in the signals is utilized and a novel method for vehicle monitoring is proposed with the help of Fourier analysis, synthesis and image processing. Methods like Doppler radar or GPS navigation are used commonly for tracking. The proposed image based approach has an added advantage that the clear image of the object (vehicle) can be used for future reference like proof of incidence, identification of owner and registration number. Keywords-Fourier Transform, dynamic texture, phase spectrum
Localization of free 3 d surfaces by the mean of photometricIAEME Publication
This document discusses a photometric stereovision technique to localize free 3D surfaces using multiple images.
It presents a photometric model linking image intensity to surface relief and reflectance properties. This model results in a system of equations with the relief variations and reflectance as unknowns. Solving the system requires 3 images under different lighting conditions.
The document applies this technique to extract the 3D relief of a corrugated plastic surface from images, demonstrating the feasibility of measuring free surfaces with the photometric stereovision approach. Accuracy of the measured surface shape is evaluated.
This document summarizes the principles of photogrammetry. It discusses the basic elements of photogrammetry including obtaining quantitative information from aerial photographs. It covers topics such as photographic scale, horizontal ground coordinates, relief displacement, exterior orientation of tilted photographs, stereoscopic vision, and the geometry of aerial stereophotographs. The purpose is to provide background information and references to support standards and guidelines for photogrammetric mapping.
Photogrammetry is the science of obtaining information about physical objects through photographs, without needing direct contact. It involves measuring and analyzing captured images. The name comes from Greek roots meaning "light", "drawing", and "to measure". Key developments included using photography for mapmaking in the 1840s-1850s, the photogrammetric stereoplotter in the 1890s, and aerial photography from balloons and planes in the 1860s-1900s, advancing the field into the digital era.
The document summarizes key concepts about aerotriangulation from a lecture on photogrammetry. It discusses that aerotriangulation establishes geometric relationships between overlapping photographs to determine supplemental ground control points. Aerotriangulation can be performed using semianalytical or analytical bundle block methods. The analytical bundle block method represents the geometric relationships between object space, camera positions, and image points in a mathematical model.
Photogrammetry is a technique for obtaining reliable spatial information about physical objects through analyzing photographs. It involves taking overlapping aerial photographs from aircraft and processing them using software to extract 3D spatial data like digital elevation models, contours, and orthophotos. The key aspects are that it is precise, cost-effective, produces 3D representations, and relies on established algorithms to extract data from overlapping photos with less manual effort than traditional surveying methods.
Method of Fracture Surface Matching Based on Mathematical StatisticsIJRESJOURNAL
ABSTRACT: Fracture surface matching is an important part of point cloud registration. In this paper, a method of fracture surface matching based on mathematical statistics is proposed. We reconstruct a coordinate system of the fractured surface points, and analyze the characteristics of the point cloud in the new coordinate system, using the theory of mathematical statistcs. The general distribution of the points is determined. The method can realize the matching relation among some point cloud.
- 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 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.
- 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.
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 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.
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 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.
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.
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.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
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.
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)
Lesson 24: Areas, Distances, the Integral (Section 041 slides)Mel Anthony Pepito
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 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 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.
- 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.
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 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.
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 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.
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.
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.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
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.
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)
Lesson 24: Areas, Distances, the Integral (Section 041 slides)Mel Anthony Pepito
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, Distances, the Integral (Section 021 slidesMel Anthony Pepito
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, Distances, the Integral (Section 021 slides)Matthew Leingang
The document outlines methods for calculating the areas of various shapes, including curved regions. It discusses the work of ancient Greek mathematicians Euclid and Archimedes, who developed early techniques for finding areas. The document also describes the work of Cavalieri in the 1600s, who introduced a new approach using rectangles inscribed in curved regions. The goal is to introduce the concept of the definite integral as a limit of Riemann sums, allowing the calculation of areas of any shape bounded by a curve.
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.
Lesson 24: Areas, Distances, the Integral (Section 021 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, 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 provides an example of using the error-tolerance game to evaluate the limit of x^2 as x approaches 0. Player 1 claims the limit is 0, and is able to show for any error level chosen by Player 2, there exists a tolerance such that the values of x^2 are within the error level when x is within the tolerance of 0, demonstrating that the limit exists and is equal to 0.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to find extreme values over a domain.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
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
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1. Section 5.1
Areas and Distances
V63.0121.002.2010Su, Calculus I
New York University
June 16, 2010
Announcements
Quiz Thursday on 4.1–4.4
. . . . . .
2. Announcements
Quiz Thursday on 4.1–4.4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 2 / 31
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 3 / 31
4. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 4 / 31
5. Easy Areas: Rectangle
Definition
The area of a rectangle with dimensions ℓ and w is the product A = ℓw.
w
.
.
.
ℓ
It may seem strange that this is a definition and not a theorem but we
have to start somewhere.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 5 / 31
6. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
.
b
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 6 / 31
7. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
.
.
b
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 6 / 31
8. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 6 / 31
9. Easy Areas: Parallelogram
By cutting and pasting, a parallelogram can be made into a rectangle.
h
.
.
b
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 6 / 31
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 6 / 31
11. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a parallelogram.
.
b
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 7 / 31
12. Easy Areas: Triangle
By copying and pasting, a triangle can be made into a parallelogram.
h
.
.
b
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 7 / 31
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 7 / 31
14. Easy Areas: Other Polygons
Any polygon can be triangulated, so its area can be found by summing
the areas of the triangles:
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 8 / 31
15. Hard Areas: Curved Regions
.
???
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 9 / 31
16. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC
– 212 BC (after Euclid)
Geometer
Weapons engineer
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 10 / 31
17. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC
– 212 BC (after Euclid)
Geometer
Weapons engineer
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 10 / 31
18. Meet the mathematician: Archimedes
Greek (Syracuse), 287 BC
– 212 BC (after Euclid)
Geometer
Weapons engineer
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 10 / 31
19. Archimedes and the Parabola
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
A=
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 11 / 31
20. Archimedes and the Parabola
1
.
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
A=1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 11 / 31
21. Archimedes and the Parabola
1
.
.1
8 .1
8
.
Archimedes found areas of a sequence of triangles inscribed in a
parabola.
1
A=1+2·
8
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 11 / 31
22. Archimedes and the Parabola
1 1
.64 .64
1
.
.1
8 .1
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 11 / 31
23. Archimedes and the Parabola
1 1
.64 .64
1
.
.1
8 .1
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 11 / 31
24. Summing the series
[label=archimedes-parabola-sum] We would then need to know the
value of the series
1 1 1
1+ + + ··· + n + ···
4 16 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 12 / 31
25. Summing the series
[label=archimedes-parabola-sum] 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 + · · · + rn ) = 1 − rn+1
So
1 − rn+1
1 + r + · · · + rn =
1−r
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 12 / 31
26. Summing the series
[label=archimedes-parabola-sum] 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 + · · · + rn ) = 1 − rn+1
So
1 − rn+1
1 + r + · · · + rn =
1−r
Therefore
1 1 1 1 − (1/4)n+1
1+ + + ··· + n =
4 16 4 1 − 1/4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 12 / 31
27. Summing the series
[label=archimedes-parabola-sum] 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 + · · · + rn ) = 1 − rn+1
So
1 − rn+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 → ∞. . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 12 / 31
28. Cavalieri
Italian,
1598–1647
Revisited the
area
problem with
a different
perspective
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 13 / 31
29. Cavalieri's method
Divide up the interval into
2
y
. =x pieces and measure the area of
the inscribed rectangles:
. .
0
. 1
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
30. Cavalieri's method
Divide up the interval into
2
y
. =x pieces and measure the area of
the inscribed rectangles:
1
L2 =
8
. . .
0
. 1 1
.
.
2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
31. Cavalieri's method
Divide up the interval into
2
y
. =x pieces and measure the area of
the inscribed rectangles:
1
L2 =
8
L3 =
. . . .
0
. 1 2 1
.
. .
3 3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
32. Cavalieri's method
Divide up the interval into
2
y
. =x pieces and measure the area of
the inscribed rectangles:
1
L2 =
8
1 4 5
L3 = + =
27 27 27
. . . .
0
. 1 2 1
.
. .
3 3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
33. Cavalieri's method
Divide up the interval into
2
y
. =x pieces 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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
34. Cavalieri's method
Divide up the interval into
2
y
. =x pieces 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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
35. Cavalieri's method
Divide up the interval into
2
y
. =x pieces 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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
36. Cavalieri's method
Divide up the interval into
2
y
. =x pieces 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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
37. Cavalieri's method
Divide up the interval into
2
y
. =x pieces 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 =?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 14 / 31
38. What is Ln ?
1
Divide the interval [0, 1] into n pieces. Then each has width .
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 15 / 31
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
( )
1 i − 1 2 (i − 1)2
· = .
n n n3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 15 / 31
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
( )
1 i − 1 2 (i − 1)2
· = .
n n n3
So
1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2
Ln = + 3 + ··· + =
n3 n n3 n3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 15 / 31
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
( )
1 i − 1 2 (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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 15 / 31
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
( )
1 i − 1 2 (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 = 3
→
6n 3
as n → ∞. . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 15 / 31
43. Cavalieri's method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 16 / 31
44. Cavalieri's method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 16 / 31
45. Cavalieri's method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
3 3
1 + 2 + 3 + · · · + (n − 1)3
=
n4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 16 / 31
46. Cavalieri's method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
3 3
1 + 2 + 3 + · · · + (n − 1)3
=
n4
The formula out of the hat is
[ ]2
1 + 23 + 33 + · · · + (n − 1)3 = 1
2 n(n − 1)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 16 / 31
47. Cavalieri's method for different functions
Try the same trick with f(x) = x3 . We have
( ) ( ) ( )
1 1 1 2 1 n−1
Ln = · f + ·f + ··· + · f
n n n n n n
1 1 1 23 1 (n − 1)3
= · 3 + · 3 + ··· + ·
n n n n n n3
3 3
1 + 2 + 3 + · · · + (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 → ∞. . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 16 / 31
48. Cavalieri's method with different heights
1 13 1 23 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
3 3 3
1 + 2 + 3 + ··· + n 3
=
n4
1 [1 ]2
= 4 2 n(n + 1)
n
n2 (n + 1)2 1
= →
4n4 4
.
as n → ∞.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 17 / 31
49. Cavalieri's method with different heights
1 13 1 23 1 n3
Rn = · 3 + · 3 + ··· + · 3
n n n n n n
3 3 3
1 + 2 + 3 + ··· + n 3
=
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 17 / 31
50. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 18 / 31
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).
For each positive integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the nth step between a and
n
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
.
. 0 . 1 . 2 . . . . i. n−1..n
xx x b b−a
x x x xn = a + n · =b
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 19 / 31
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).
For each positive integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the nth step between a and
n
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
.
. 0 . 1 . 2 . . . . i. n−1..n
xx x b b−a
x x x xn = a + n · =b
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 19 / 31
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).
For each positive integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the nth step between a and
n
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
.
. 0 . 1 . 2 . . . . i. n−1..n
xx x b b−a
x x x xn = a + n · =b
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 19 / 31
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).
For each positive integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the nth step between a and
n
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
.
. 0 . 1 . 2 . . . . i. n−1..n
xx x b b−a
x x x xn = a + n · =b
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 19 / 31
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).
For each positive integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the nth step between a and
n
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
.
. 0 . 1 . 2 . . . . i. n−1..n
xx x b b−a
x x x xn = a + n · =b
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 19 / 31
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 20 / 31
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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 20 / 31
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
. ..1
xb
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
. x
.1 ..2
xb
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
. x
.1 x
.2 ..3
xb
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
. x
.1 x
.2 x
.3 ..4
xb
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 b 5
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 b 6
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 . . . . . . . .
ax x x x x x x
. . . . . . . ..
matter what choice of ci we 1 2 3 4 5 6 b 7
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
. . . . . . . . ..
b
matter what choice of ci we 1 2 3 4 5 6 7 8
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
. . . . . . . . . ..
b
matter what choice of ci we 1 2 3 4 5 6 7 8 9
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 . . . . . . . . . . .
ax 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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 xx xb
. . . . . . . . . . . ..
matter what choice of ci we 1 2 3 4 5 6 7 8 9 1011
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 10 12
11
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 . ...............
a xxxxxxxxxxxxxxb
.. . . . . . . . .. . . . . . .
x1 2 3 4 5 6 7 8 910 12 14
matter what choice of ci we 11 13 15
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 . ................
a xxxxxxxx xxxxxxb
. . . . . . . . . .. . . . . . . .
x1 2 3 4 5 6 7 8x10 12 14 16
matter what choice of ci we 9 11 13 15
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 xxxxxxxx xxxxxxxb
.. . . . . . . . .. . . . . . . . .
x1 2 3 4 5 6 7 8x10 12 14 16
matter what choice of ci we 9 11 13 15 17
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
.. . . . . . . . .. . . . . . . . . .
x12345678910 12 14 16 18
x 11 13 15 17
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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
.. . . . . . . . .. . . . . . . . . . .
x1234567891012141618
x 1113151719
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
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 . ....................
axxxxxxxx xxxxxxxxxxb
.. . . . . . . . .. . . . . . . . . . . .
x123456789 1113151719
x101214161820
matter what choice of ci we
made.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 21 / 31
78. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
79. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4)
Want the slope of a curve
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
80. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
81. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of
lines
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
82. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
83. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons
Approximate curve with a
line
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
84. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area of
lines polygons
Approximate curve with a Approximate region with
line polygons
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
85. Analogies
The Tangent Problem The Area Problem (Ch. 5)
(Ch. 2–4) Want the area of a curved
Want the slope of a curve region
Only know the slope of Only know the area 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
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 22 / 31
86. Outline
Area through the Centuries
Euclid
Archimedes
Cavalieri
Generalizing Cavalieri’s method
Analogies
Distances
Other applications
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 23 / 31
87. Distances
Just like area = length × width, we have
distance = rate × time.
So here is another use for Riemann sums.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 24 / 31
88. Application: Dead Reckoning
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 25 / 31
89. Computing position by Dead Reckoning
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.1 Areas and Distances June 16, 2010 26 / 31
90. Solution
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.
. . . . . .
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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.
. . . . . .
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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
. . . . . .
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93. Surplus by picture
c
. onsumer surplus
p
. rice (p)
s
. upply
.∗ .
p . . quilibrium
e
d
. emand f(q)
. .
.∗
q q
. uantity (q)
. . . . . .
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94. 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.
. . . . . .
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