Answer
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} {x^2}y{\rm{d}}A = \frac{4}{3}$
Work Step by Step
We have $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} {x^2}y{\rm{d}}A$ and ${\cal R} = \left[ { - 1,1} \right] \times \left[ {0,2} \right]$.
Write
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} {x^2}y{\rm{d}}A = \mathop \smallint \limits_{x = - 1}^1 \left( {\mathop \smallint \limits_{y = 0}^2 {x^2}y{\rm{d}}y} \right){\rm{d}}x$
$ = \mathop \smallint \limits_{x = - 1}^1 {x^2}\left( {\frac{1}{2}{y^2}|_0^2} \right){\rm{d}}x$
$ = 2\mathop \smallint \limits_{x = - 1}^1 {x^2}{\rm{d}}x$
$ = 2\left( {\frac{1}{3}{x^3}|_{ - 1}^1} \right)$
$ = 2\left( {\frac{1}{3} + \frac{1}{3}} \right)$
$ = \frac{4}{3}$
