Answer
$$\frac{15 \pi}{2}$$
Work Step by Step
State representative radius and height of the cylinder
Representative radius is some value of x over the interval $1 \leq x \leq 2$
Cylinder height is $x^{2}=y$
integrate with respect to $x$
$$
\begin{array}{l}
2 \pi \int_{1}^{2} x\left(x^{2}\right) d x \\
2 \pi \int_{1}^{2} x^{3} d x \\
2 \pi\left[\frac{x^{4}}{4}\right]_{1}^{2} \\
\left[\frac{2^{4}}{4}-\frac{1^{4}}{4}\right]2 \pi \\
\left[\frac{16}{4}-\frac{1}{4}\right]2 \pi=\left[\frac{15}{4}\right]2 \pi=\frac{15 \pi}{2}
\end{array}
$$
