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?