Question

Find the volume of the object by rotating the region bounded by the curves f(x) = e^x^2 and y = 0 and x ∈ [0, sqrt(ln 2)] around the y-axis.

Answer 1

Using the method of shells, you have
\[V=2\pi\int_0^{\sqrt{\ln2}}x e^{x^2}\,\mathrm{d}x\]The integral can be computed with a simple change of variables.

I would normally draw a picture for this kind of problem, but I'm on a laptop and the mousepad is being rather finicky these days. Sorry for the inconvenience. On the bright side, W|A has a neat feature that can do this for you (though it's kind of buggy at times, so be careful if you intend to use it in the future): http://www.wolframalpha.com/input/?i=revolve+region+bounded+by+e%5E(x%5E2)+and+y%3D0+between+x%3D0+and+x%3Dsqrt(ln2)+about+y+axis&dataset=