Take the curve \(f(x) = x^x\) and rotate it around the line \(g(x) = x\).

How much volume does that enclose?

Well, it's infinite. The curve touches the line once and pulls away forever. It's like a trumpet, only you couldn't blow it because the hole is closed and you'd never hear it anyway since the horn goes on forever.

What if we cut off the upper section after a few units? Then we can measure it. It's still a really fucky shape though. We could maybe approximate it with a cone, but we're gonna have to use integrals to get an accurate result.