The region bounded by y = 1/(x^2), y = x and x = 2 is rotated around the vertical axis x = -1. Using vertical strips and x integration, what is the volume of the solid formed?

I was able to sketch the region and find its area (its area is 1, right?) but I'm having trouble finding the volume. I think I have to use the shell method, but I'm not sure how to approach the volume calculation.
On a quick attempt, I got 2PI(5/6 - ln2) but I'm very doubtful of this answer.

Question

The region bounded by y = 1/(x^2), y = x and x = 2 is rotated around the vertical axis x = -1. Using vertical strips and x integration, what is the volume of the solid formed?

I was able to sketch the region and find its area (its area is 1, right?) but I'm having trouble finding the volume. I think I have to use the shell method, but I'm not sure how to approach the volume calculation. 
On a quick attempt, I got 2PI(5/6 - ln2) but I'm very doubtful of this answer.

mathmale · Answer

Sketching the region in question was a good idea.  Next time, share your sketch.  Note that because you're looking for the volume of a solid, there's no point in finding the area of the region.

Here is a quick sketch of the region in question:|dw:1408852875520:dw|