# Thread: Volume of a Solid: Region Bound and Rotated about y-axis

1. ## Volume of a Solid: Region Bound and Rotated about y-axis

The region bounded by $y=e^{-x^2},\ y=0,\ x=0,$ and $x=1$ is revolved about the y-axis. Find the volume of the resulting solid.

I know that the answer is $2\pi\int_0^1xe^{-x^2}$, however I am not sure how to get there. I drew a picture, and I can't tell if I'm supposed to approximate with disks or shells...or approach it a different way. Any insight offered is appreciated. Thanks in advance.

2. It looks like wherever you found the answer used the method of shells. The formula for shell integration is $\displaystyle 2 \pi \int_a^b x f(x) \, dx$ (when the axis of revolution is the y-axis).

According to the formula for shell integration, the volume V of the solid is given by: $\displaystyle 2 \pi \int_0^1 x e^{-x^2} \, dx$