So I'm suppose to find the volume of revolution of an area enclosed by

and

rotated about x=1.

So I'm using the cylindrical shell method

and I'm integrating with respect to x; finding the integration points for the limits of integration, I get b=1 or a=-1.

Since the axis is x=1, then r=1-x. The height,

Therefore, the definite integral is (factoring out the 2pi from the start and 2 from the height):

.

Since it's symmetrical I can change the limits to 0 and 1, multiplying the integral by a factor of 2. Then expanding yields:

By FTC:

evaluated from 0 to 1.

That yields

and simplifying gets:

However, I'm apparently wrong as the answer key says that it should be

. I can't see my mistake, anyone see where I went wrong?