Answer
$\dfrac{2336}{27}$
Work Step by Step
The volume under the surface is given by :$z=f(x,y)$ and above the region $D$ in the xy-plane can be expressed as: $\ Volume ; V=\iint_{D} f(x,y) \space dA$
Our aim is to calculate the volume of the given surface.
The domain $D$ can be expressed as follows:
$D=\left\{ (x, y) | y^2 \leq x \leq 4 , \ -2 \leq y \leq 2 \right\}
$
Now, $V =\iint_{D} f(x,y) \ dA \\=\int_{-2}^{2} \int_{y^2}^{4} 1+x^2y^2 \ dx \ dy \\ =\int_{-2}^{2} [x+\dfrac{x^3y^2}{3}]_{y^2}^{4} \ dx \\= \int_{-2}^{2} [ 4+\dfrac{64y^2}{3}]-[y+\dfrac{y^8}{3}] \ dy \\ = 4[y]_{-2}^{2} +\dfrac{61}{9} [y^3]_{-2}^{2} -\dfrac{1}{27}\times [y^9]_{-2}^{2} \\= \\= 2[8+\dfrac{488}{9}-\dfrac{512}{27}] \\= \dfrac{2336}{27}$
