My favorite is the rectangular hyperbola revolved around one of
its axes. The volume integral converges for this beast, but the
surface area integraldoes not. Thus one can fill it with a finite
volume of paint, but one requires an infinite amount of paint to
cover the inside surface!
Yesterday I had to integrate the Planckian distribution to find
the number density of photons in a blackbody radiation field*.
Maple refused to do the integral. Abramowitz and Stegun struck
out on it. http://www.integrals.com was no help.
Guess what. I found an excellent treatment on the web! You might
want to take a look at http://www.astro.virginia.edu/~eww6n . I
have found an error or two in what is here, but I'm impressed.
Leigh
*Yes, I know it's a standard problem in undergrad physics - I had
assigned it to my class, and I had forgot how to do it.