Answer
$\frac{11}{70}$
Work Step by Step
Volume under the function $ f (x, y)=z $ and over the region $ R $ is given by
\[
\begin{array}{c}
\iint_{R} f(x, y) d A=V \\
V=\iint_{R} x^{2}+3 y^{2} d A
\end{array}
\]
$=\int_{0}^{1} \int_{x^{2}}^{x} x^{2}+3 y^{2} d y d x$
$=\int_{0}^{1}\left[x^{2} y+y^{3}\right]_{y=x^{2}}^{y=x} d x$
$=\int_{0}^{1}\left[x^{3}+x^{3}\right]-\left[x^{4}+x^{6}\right] d x$
$=\int_{0}^{1} -x^{6}-x^{4} +2 x^{3}d x$
$=\left[\frac{x^{4}}{2}-\frac{x^{5}}{5}-\frac{x^{7}}{7}\right]_{0}^{1}$
$=\frac{1}{2}-\frac{1}{5}-\frac{1}{7}$
$=\frac{11}{70}$
