\[\textrm{volume}\ =\ \int_a^b A(x)dx\] where \(A(x)\) is the cross section area of the slice at \(x\).

See the example in class.

\[\textrm{volume}\ =\ \int_a^b 2\pi x f(x) dx\] for the volume of the solid of revolution obtained by rotating the area under the graph of \(y=f(x) \ge 0\) about the \(y\)-axis.

For example, \(y=e^{-x^2}\), \(0\le x\le 1\).
Then \[V\ =\ \int_0^1 2\pi x e^{-x^2}dx\] \[=\Big[ - \pi e^{-x^2}\Big]_0^1 = \pi(1-e^{-1}) \]