The radius should be "x - 1" .
The region is revolving around the line x= 1. The radius is distance from this line. So radius varies from 0 to 1 as x goes from 1 to 2.
Therefore , r = x - 1.
Everything else is correct.
Here is the integral solution:
[tex]V = 2\pi \int_1^2 (x-1) x^2 dx \\ \\ = 2\pi \int_1^2 x^3 -x^2 dx \\ \\ = 2\pi |_1^2 (\frac{x^4}{4} - \frac{x^3}{3}) \\ \\ =2\pi (\frac{4}{3} - (-\frac{1}{12})) \\ \\ =\frac{17\pi}{6} [/tex]
