Module 5: Section 4, Part 2
The Definite Integral

Introduction

In the last section we were introduced to the definite integral and the fundamental theorem of calculus which gives us a way to evaluate the definite integral. In this section we will discuss the concept of the definite integral more in depth, look at some applications and theorems involving the definite integral, and apply our antiderivative techniques to evaluate definite integrals.

The Definite Integral

First, let's recall what we get from the definite integral:

Click here if you would like to see a short video of a graphical representation of the definition of the definite integral.

The Fundamental Theorem of Calculus

Ok, now that we have those reminders, let's evaluate a few definite integrals.

Calculating Definite Integrals

Recall that the fundamental theorem of calculus says that we can calculate a definite integral if we can find the antiderivative of the integrand.

Let's take the following example step by step.

Examples.

Example 1.

Example 2.

Try to calculate the following definite integrals, and then few the LiveScribe videos to check your answers:

LiveScribe Solution PDF Version

Video Examples of Calculating Definite Integrals

Video Example 1.

In this video the instructor calculates a basic definite integral. Follow this link.

Video Example 2.

In this video the instructor calculates a definite integral using u-substitution. Follow this link.

More on Definite Integrals

Definite Integrals Versus Indefinite Integrals

Now if we have definite integrals, we also have indefinite integrals (or general antiderivatives). Let's take a look at how the characteristics of the two integrals differ:

Theorems Involving Definite Integrals

We took a quick look at two theorems in the last section. Let's revisit those, but this time for definite integrals.

1. This theorem is saying that we can break up a definite integral over addition or subtraction and calculate the definite integral term by term.
2. This theorem says we can factor a constant out of the definite integral, calculate the definite integral, then multiply the result by that constant.

1. This theorem says that if we switch our limits of integration, that changes the sign of our definite integral.
2. This theorem says we can break up our input interval at some point c, calculate the two definite integrals, then determine the sum of the definite integrals to determine the value of the whole integral.

1. This theorem says that if we are taking the definite integral of a constant, we can multiply the constant by the length of the interval.
2. This theorem says that if both the limits of integration are the same, the definite integral will be 0. 

The following video discusses why these two theorems are true from a grpahical standpoint.

If you would like to see proofs of some of these properties, follow this link.

Check Your Understanding

Use the following two definite integrals to answer the questions

Applications of the Definite Integral

We saw in the last section that if we integrate a velocity function over an interval of time, we get the distance traveled over that interval. Let's look at a couple other scenarios.

Example.

Listen to the description here:

Check Your Understanding