Answer
$$170$$
Work Step by Step
Volume under the function $f(x, y)=z$ and above the region $R$ is given by
\[
\begin{array}{c}
\iint_{R} f(x, y) d A=V \\
\iint_{R} y^{2}+9 x^{2} d A=V \\
=\int_{0}^{3} \int_{0}^{2} 9 x^{2}+y^{2} d y d x \\
=\int_{0}^{3}\left[9 y x^{2} +\frac{y^{3}}{3}\right]_{0}^{2} d x \\
=\int_{0}^{3} \frac{8}{3}+ 18 x^{2} d x
\end{array}
\]
$$=[\frac{8 x}{3}+6 x^{3} ]_{0}^{3}$$
\[
170=\frac{8 \cdot 3}{3}+6 \cdot 3^{3}
\]
