18.01SC | Fall 2010 | Undergraduate

Single Variable Calculus

3. The Definite Integral and its Applications

Part A: Definition of the Definite Integral and First Fundamental Theorem

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The definite integral of a function describes the area between the graph of that function and the horizontal axis. The First Fundamental Theorem of Calculus confirms that we can use what we learned about derivatives to quickly calculate this area.

» Session 43: Definite Integrals
» Session 44: Adding Areas of Rectangles
» Session 45: Some Easy Integrals
» Session 46: Riemann Sums
» Session 47: Introduction of the Fundamental Theorem of Calculus
» Session 48: The Fundamental Theorem of Calculus
» Session 49: Applications of the Fundamental Theorem of Calculus
» Session 50: Combining the Fundamental Theorem and the Mean Value Theorem
» Problem Set 6

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Overview

In this session you will:

  • Do practice problems
  • Use the solutions to check your work

Problem Set

Use Integration (PDF) to do the problems below.

Section Topic Exercises
3B Definite integrals 2a, 2b, 3b, 4a, 5

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

Section Topic Exercises
4J Other applications 1 (Set up integral, but don’t evaluate)

Use Integration (PDF) to do the problems below.

Section Topic Exercises
3C Fundamental theorem of calculus 1, 2a, 3a, 5a
3E Change of variables; Estimating integrals 6b, 6c

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

Section Topic Exercises
4J Other applications 2

Solutions

Solutions to Integration problems (PDF)

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

By dividing a region up into thinner and thinner rectangles, we can describe its area as a limit of sums of rectangle areas. Even when using summation notation, this process can be time consuming.

Lecture Video and Notes

Video Excerpts

Clip 1: Example: f(x)= x2

Clip 2: Summation Notation

Recitation Video

Summation Notation Practice

Worked Example

Summation

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Overview

When we found the area under the graph of y=x^2 we used a Riemann sum. These sums of rectangle areas can easily be translated into integrals by allowing the rectangles to become infinitesimally thin.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Riemann Sums

Worked Example

Riemann Sum Practice

Recitation Video

Riemann Sum

Lecture Video and Notes

Video Excerpts

Clip 1: Example: Cumulative Debts

Recitation Video

Computing the Volume of a Paraboloid

Diffusion of a Chemical

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Overview

In this session we learn how to use antiderivatives to calculate the value of a definite integral.

Lecture Video and Notes

Video Excerpts

Clip 1: The First Fundamental Theorem of Calculus

Clip 2: Using the First Fundamental Theorem

Recitation Video

Definite Integrals of tan (x)

Worked Example

Practice With Definite Integrals

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Overview

Definite integrals “sum up” infinitesimal changes; one such change is adding the area of an infinitesimal rectangle. Another is traveling an infinitesimal distance forward. In this session we use definite integrals to compute distance traveled over time given a velocity function. We discuss negative integrands and then list a few properties of definite integrals.

Lecture Video and Notes

Video Excerpts

Clip 1: Interpretation of the Fundamental Theorem

Clip 2: The Fundamental Theorem and Negative Integrands

Clip 3: Properties of Integrals

Worked Example

Integral of sin(x) + cos(x)

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Overview

In this session we see how definite integrals can be used in estimation, to find upper or lower bounds on an answer. Then we examine how the limits on the definite integral interact with substitutions.

Lecture Video and Notes

Video Excerpts

Clip 1: Example of Estimation

Clip 2: Example: Change of Variables

Recitation Video

Definite Integral by Substitution

Worked Example

Integration by Change of Variables

Lecture Video and Notes

Video Excerpts

Clip 1: Substitution When u’ Changes Sign

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Course Info

Instructor
Departments
As Taught In
Fall 2010
Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos