Answer
172
Work Step by Step
We find:
\[
\begin{array}{c}
\iint_{R} f(x, y) d A =V\\
V=\iint_{R} 3 x^{3}+3 x^{2} y d A \\
=\int_{1}^{3} \int_{0}^{2} 3 x^{3}+3 x^{2} y d y d x \\
=\int_{1}^{3}\left[3 x^{3} y+\frac{3}{2} x^{2} y^{2}\right]_{0}^{2} d x \\
=\int_{1}^{3} 6 x^{3}+6 x^{2} d x \\
=\left[\frac{3}{2} x^{4}+2 x^{3}\right]_{1}^{3} \\
=\left[\frac{243}{2}+54\right]-\left[\frac{3}{2}+2\right]=172
\end{array}
\]
