Badly Behaved Functions Examples

Determine whether

converges or diverges. If it converges, find its value.

By looking at the function or its graph we can see that  is badly behaved at the lower limit of integration, x = 0. This means when we rewrite the integral as a limit we need to change the lower limit of integration.

We know how to work out this integral.

Since ln b approaches -∞ as b approaches 0 from the right, this limit does not exist. The integral

diverges. The area of this region is infinite.

As we mentioned before, you should be careful with your notation. For these improper integrals, we have one thing to add to that discussion. When we write out the improper integrals as limits, we need to use one-sided limits. In the case of the integral

we had a one-sided limit as b approached 0 from the right. If the function were badly behaved at the right endpoint of integration, we would have a one-sided limit with b approaching the endpoint from the left.

Determine whether

converges or diverges. If it converges, find its value.

On this interval,  is badly behaved at the upper limit of integration. This means when we rewrite the integral as a limit we need to change the upper limit of integration, and use a one-sided limit with b approaching 0 from the left.

We work out the integral as before:

This limit diverges, and so the original integral diverges also. This area is infinite.

Determine whether

converges or diverges.

This function has an asymptote at 0: This means we have to split up the improper integral:

Since we already know that

diverges, the original integral diverges also.

Here's the wrong way to find

We pretend it's a normal definite integral:

We'll say again: this is wrong. This is not a definite integral; it's a weird improper integral that we can't evaluate without doing weird limit stuff.

Before doing any integral, graph your function and check for bad behavior!