I think my biggest problem here is I can not find a good way to find the square root in this problem
$x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2} \ \ \ \ 1 \le x \le 2$
$\int_1^2 2 \pi \cdot {\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}} \sqrt{1 + (1/3 (y^2 + 2)^{\frac{3}{2}})^2}$
$\int_1^2 2 \pi \cdot \frac{1}{3}\left(y^2+2\right)^\frac{3}{2} \sqrt{1 + y^3 + 2y}$
Here is where I am stuck, I have tried u-substitution and trig and I can not make this work. I have two pages of notes but I expect that they would be useless to type up.