Determine whether the integral converges or diverges. Indicate on a graph the region whose weighted area is given by the integral. If the integral converges, find its value.
First split up the integral into two improper integrals with only one badly-behaved limit each:
Now work out each of those improper integrals.
Since c is approaching -∞, the quantity ec is approaching 0. So
Now for the other integral.
This limit diverges. Since one of the two improper integrals diverges, the original integral diverges also.
If we had chosen to evaluate the integral
first, we wouldn't have had to bother with the other one at all. If we had only drawn a picture of the function first, we could have seen from the graph that
would diverge! The moral of the story: look at the graph of the function before you start integrating.
If you did look at a graph first and did this the more efficient way, give yourself a pat on the back. Or a carrot.