Part C: Average Value, Probability and Numerical Integration

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

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3. The Definite Integral and its Applications

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Calculating averages and probabilities is an important application of integration. Although we can differentiate almost any function we can describe, there are functions whose integral we cannot directly describe; this part of the course describes numerical methods used to calculate the values of such integrals.

» Session 60: Integrals and Averages
» Session 61: Integrals and Weighted Averages
» Session 62: Integrals and Probability
» Session 63: Numerical Integration
» Session 64: Numerical Integration, Continued
» Session 65: Bell Curve, Conclusion
» Problem Set 8

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Overview

In this session you will:

Do practice problems
Use the solutions to check your work

Problems Set

Use Applications of Integration (PDF) to do the problems below.

Section	Topic	Exercises
4D	Average value	2, 3, 5

Solutions

Solutions to Applications of Integration problems (PDF)

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

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Overview

This session explains how definite integrals can be used to calculate the average value of a function on an interval, then presents several examples.

Lecture Video and Notes

Video Excerpts

Clip 1: Average Value

Clip 2: Average Height (3 min)

Clip 3: Average Height with Respect to Arc Length

Recitation Video

Average Velocity

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Worked Example

Average Bank Balance

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Overview

We use a “weighted average” to take into account differences in energy, value, density, etc. of the region we’re integrating over.

Lecture Video and Notes

Video Excerpts

Clip 1: Weighted Averages

Clip 2: Boiling Cauldron: Introduction

Clip 3: Boiling Cauldron, Continued

Worked Example

Weighted Average

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Overview

Calculating probabilities is an important application of integration. In these examples we use integrals to find the probability of an event occurring in a specified region, as opposed to outside that region.

Lecture Video and Notes

Video Excerpts

Clip 1: Probability Example

Recitation Video

Average x-Coordinate in a Region

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Lecture Video and Notes

Video Excerpts

Clip 1: Probability Summary

Clip 2: Errata: Heat is Energy

Clip 3: Example: Boy Near a Dart Board

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Overview

Sometimes it’s not practical to find the value of a definite integral by using an antiderivative. This session presents three different numerical methods of approximating the value of a definite integral.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Numerical Integration

Clip 2: Riemann Sums

Clip 3: Trapezoidal Rule

Clip 4: Simpson’s Rule

Recitation Video

Explanation of Simpson’s Rule

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Using the Trapezoid and Simpson’s Rules

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Worked Example

Numerical Integration

Please use the mathlet below to complete this problem.

Mathlet

Riemann Sums

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Overview

This session shows that Simpson’s rule is superior to the trapezoidal rule for a sample integral of 1/x. After that, Professor Jerison shares hints for remembering the two rules.

Lecture Video and Notes

Video Excerpts

Clip 1: Integral of 1/x, Trapezoids

Clip 2: Integral of 1/x, Simpson’s rule

Clip 3: Numerical Integration Study Tips

Worked Example

The Basis for Simpson’s Rule

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Overview

In this session we use a clever trick involving finding volumes by slices to calculate the area under the bell curve, neatly avoiding the problem of finding an antiderivative for e^{-x^2}.