Answer
$$\frac{8}{15}$$
Work Step by Step
Volume under the function $ f (x, y)=z $ and over the region $ R $ is given by
\[
\begin{array}{c}
\iint_{R} -x +1d A=V \\
=\int_{-1}^{1} \int_{y^{2}}^{1} 1-x d x d y \\
=\int_{-1}^{1}\left[x-\frac{x^{2}}{2}\right]_{y^{2}}^{1} d y \\
=\int_{-1}^{1}\left[1-\frac{1^{2}}{2}\right]-\left[y^{2}-\frac{\left(y^{2}\right)^{2}}{2}\right] d y \\
=\int_{-1}^{1} \frac{1}{2}-y^{2}+\frac{y^{4}}{2} d y \\
=\left[\frac{y}{2}-\frac{y^{3}}{3}+\frac{y^{5}}{10}\right]_{-1}^{1} \\
\frac{8}{15}=\left[-\frac{1}{3}+\frac{1}{10}+\frac{1}{2}\right]2
\end{array}
\]
