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?
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 ?
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.