Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_{ - 2}^2 {\int_{{x^2}}^{8 - {x^2}} {xdydx} } \cr
& {\text{Integrate with respect to }}y \cr
& = \int_{ - 2}^2 {\left[ {xy} \right]_{{x^2}}^{8 - {x^2}}} dx \cr
& = \int_{ - 2}^2 {\left[ {x\left( {8 - {x^2}} \right) - x\left( {{x^2}} \right)} \right]} dx \cr
& = \int_{ - 2}^2 {\left( {8x - {x^3} - {x^3}} \right)} dx \cr
& = \int_{ - 2}^2 {\left( {8x - 2{x^3}} \right)} dx \cr
& {\text{Integrate with respect to }}x \cr
& = \left[ {4{x^2} - \frac{1}{2}{x^4}} \right]_{ - 2}^2 \cr
& {\text{Evaluating}} \cr
& = \left[ {4{{\left( 2 \right)}^2} - \frac{1}{2}{{\left( 2 \right)}^4}} \right] - \left[ {4{{\left( { - 2} \right)}^2} - \frac{1}{2}{{\left( { - 2} \right)}^4}} \right] \cr
& = \left[ {16 - 8} \right] - \left[ {16 - 8} \right] \cr
& = 0 \cr} $$
