Part B: Partial Fractions, Integration by Parts, Arc Length, and Surface Area

Course Info

Instructor

Prof. David Jerison

Departments

Mathematics

As Taught In

Fall 2010

Level

Undergraduate

Topics

Mathematics
- Calculus
- Differential Equations

Learning Resource Types

Exams with Solutions

Lecture Notes

Lecture Videos

Problem Sets with Solutions

Simulations

Recitation Videos

Download Course

4. Techniques of Integration

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Partial Fractions Decomposition and Integration by Parts are techniques for simplifying complex integrals. In this part of the course we also describe how to use integration to find the length of a portion of a graph and the surface area of a rotationally symmetric surface.

» Session 74: Integration by Partial Fractions
» Session 75: Advanced Partial Fractions
» Session 76: Integration by Parts
» Session 77: Volume of a Wine Glass
» Session 78: Computing the Length of a Curve
» Session 79: Surface Area
» Problem Set 10

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Overview

In this session you will:

Do practice problems
Use the solutions to check your work

Problem Set

Use Integration Techniques (PDF) to do the problems below.

Section	Topic	Exercises
5E	Integration by partial fractions	2, 3, 5, 6, 10h (complete the square)
5F	Integration by parts; Reduction formulas	1a, 2d, then 2b, 3

Solutions

Solutions to Integration Techniques problems (PDF)

This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.

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Overview

If guessing and substitution don’t work, we can use the method of partial fractions to integrate rational functions. This session presents the time saving “cover-up method” for performing partial fractions decompositions.

Lecture Video and Notes

Video Excerpts

Clip 1: Partial Fractions I

Clip 2: Introduction to the Cover-Up Method

Clip 3: The Cover-Up Method

Worked Example

Integrate by Partial Fractions

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Overview

Last session we learned to use partial fractions to integrate rational functions for which the degree of the numerator was less than the degree of the denominator, and where the denominator had particularly nice factors. In this session we learn how to use partial fractions under more adverse conditions.

Lecture Video and Notes

Video Excerpts

Clip 1: Repeated Factors

Clip 2: Quadratic Factors

Worked Example

Integral of (x-11)/(x²+9)(x+2)

Lecture Video and Notes

Video Excerpts

Clip 1: Long Division

Worked Example

Integral of x³/x²-1

Recitation Video

Partial Fractions Decomposition

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Lecture Video and Notes

Video Excerpts

Clip 1: Partial Fractions – Big Example

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Overview

Integration by parts is useful when the integrand is the product of an “easy” function and a “hard” one. In this session we see several applications of this technique; note that we may need to apply it more than once to get the answer we need.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Integration by Parts

Recitation Video

Finding u and v’ When Integrating by Parts

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Lecture Video and Notes

Video Excerpts

Clip 1: Integral of Natural Log

Clip 2: Integral of ln(x)2

Clip 3: A Reduction Formula

Clip 4: Another Reduction Formula

Recitation Video

Integrating sinⁿ(x) Using Reduction

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Worked Example

Integral of x⁴ cos(x)

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Overview

Using results from the previous session, we calculate the volume of a solid of revolution in two different ways.

Lecture Video and Notes

Video Excerpts

Clip 1: By Horizontal Slices

Clip 2: By Vertical Slices

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Overview

To find the length of a curve we break it up into infinitesimal pieces. Each piece is approximately a straight line segment; we use differential notation to compute and sum the lengths of these pieces.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Arc Length

Clip 2: Example: y=mx

Clip 3: Example: Circular Arc

Clip 4: Example: Length of a Parabola

Recitation Video

Arc Length of y=x^3/2

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Worked Example

Arc Length of y=ln(x)

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Overview

To find the surface area of a curve revolved around an axis, we break the curve into infinitesimal segments ds then sum up the areas of the bands formed by rotating each segment ds about the axis.