# Review of Improper integrals

First, let's review the meaning of the integral. In the integral what does the represent ?

In the integral what does the represent ?

Explain why the integral is *improper.*

What's another example of an improper integral?

**separating into two (or more) integrals and evaluating left-hand or right-hand limits.**We can evaluate indefinite integrals that involve unbounded regions of integration

**by finding limits at infinity**.

Let's try an example. Evaluate We split the integral into the sum of two integrals: Now each integral only has a discontinuity on the boundary (t=2). If we can find an antiderivative of , we can apply the Fundamental Theorem of Calculus, taking the left or right hand limit when applying to the endpoint t=2. What is an antiderivative of ?

To evaluate , you find the limit of the antiderivative as approaches 2 from the left, the value of the antiderivative at t=0, and subtract them. What is ?

To you evaluate , you find value of the antiderivative at t=5, the limit of the antiderivative as approaches 2 from the right, and subtract them. What is ?

What can you conclude about ? Explain.

Most of the improper integrals we will encounter in this class will be a different type. Explain why the integral is improper.

What is an antiderivative of ?

To evaluate the integral, you take the limit of the antiderivative as approaches , and then subtract the value of the antiderivative at What is the value of ?

*t*(upper limit of integration) in the integrals below. Then answer the questions.

Describe how changes as increases.

Contrast with how changes as increases.

Suppose the function and for all . What can you conclude about ?

Suppose the function and for all . What can you conclude about ?

Justify your reasoning for the last two questions.