Review of Improper integrals

Author:
Steve, Yin Su
In this activity, we will review the concept 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?

Improper Integrals can also be formed if the integrand is discontinuous on the region of integration. For example, is improper for because the function is undefined (and therefore discontinuous) at .
The Fundamental Theorem of Calculus does not hold (in general) for Improper Integrals. That is, we cannot simply evaluate where is an antiderivative of . If we can find an antiderivative, we can evaluate indefinite integrals that involve discontinuities by 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 ?

Select all that apply
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Check my answer (3)
Manipulate the slider to vary 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 ?

Select all that apply
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  • E
Check my answer (3)

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

Select all that apply
  • A
  • B
  • C
  • D
  • E
Check my answer (3)

Justify your reasoning for the last two questions.