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.

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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

We can use antiderivatives to find the area bounded by some vertical line x=a, the graph of a function, the line x=b, and the x-axis. We can prove that this works by dividing that area up into infinitesimally thin rectangles.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction to Definite Integrals

Clip 2: Definition of Definite Integrals

### Worked Example

Integration Intuition

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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

### Worked Example

Summation

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### Overview

We now know how to find the area under the graph of y=x^2. We can easily find the areas under the graphs of y=x and y=1. This allows us to guess a formula for the area under the graph of y=x^n.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Simple Definite Integrals

Clip 2: Summary of Examples

### Worked Example

Integral of |x|

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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

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Example: Cumulative Debts

### Recitation Video

#### 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

### 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

### Worked Example

Integration by Change of Variables

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Substitution When u’ Changes Sign

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### Overview

We’ve seen how definite integrals and the mean value theorem can be used to prove inequalities. In this session we see that FTC1 renders MVT obsolete and review an exam problem on the MVT.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Review of the Fundamental Theorem

Clip 2: The Fundamental Theorem and the Mean Value Theorem

Clip 3: The Mean Value Theorem and Estimation

### Worked Example

Estimating ln(5)

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## Course Info

Fall 2010
##### Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos