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. |