Answer
$961.181$
Work Step by Step
We have: $f(x,y) =x^3y^4+xy^2$
The volume under the surface $z=f(x,y)$ and above the region $D$ in the $xy$-plane can be expressed as follows: $$\ Volume =\iint_{D} f(x,y) \ dA$$
and the region $D$ using the point of intersection can be expressed as follows:
$$D=\left\{ (x, y) | 0 \leq x \leq 2, \ x^2+x \geq y \geq x^3-x \right\}$$
Apply the polar co-ordinates system $x= r \cos \theta$ and $ y= r \sin \theta$
Now, $\ Volume =\iint_{D} f(x,y) \ dA\\=\iint_{D} x^3y^4+xy^2 \ dA = \int_{0}^{2} \int_{x^3-x}^{x^2+x} (x^3y^4+xy^2) dy dx $
The integral can be computed by using a calculator as follows:
$\ Volume ; V=\int_{0}^{2} \ \int_{x^3-x}^{x^2+x} \ (x^3y^4+xy^2) \ dy \ dx \\ \approx 961.181$
