Question

find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis.

y=e^x, y=0, x=0, x=ln3

Answer 1

the answer on the back of the book says\[4\pi\]

Answer 2

\[f(x)=e^x\]
\[V=\pi\int\limits_{0}^{\ln{3}}f^2(x)dx=\pi\int\limits_{0}^{\ln{3}}e^{2x}dx=\frac{\pi}{2}e^{2x}|_{0}^{\ln{3}}=\frac{\pi}{2}(e^{2\ln{3}}-1)=\]
\[=\frac{\pi}{2}(9-1)=4\pi\]