Answer
8
Work Step by Step
$$\eqalign{
& \int_{ - 2}^2 {\int_1^2 {\int_1^e {\frac{{x{y^2}}}{z}} dz} dx} dy \cr
& = \int_{ - 2}^2 {\int_1^2 {x{y^2}\left[ {\int_1^e {\frac{1}{z}} dz} \right]} dx} dy \cr
& {\text{Integrate with respect to }}z \cr
& = \int_{ - 2}^2 {\int_1^2 {x{y^2}\left[ {\ln \left| z \right|} \right]_1^e} dx} dy \cr
& = \int_{ - 2}^2 {\int_1^2 {x{y^2}\left( 1 \right)} dx} dy \cr
& = \int_{ - 2}^2 {\int_1^2 {x{y^2}} dx} dy \cr
& = \int_{ - 2}^2 {\left[ {\frac{{{x^2}{y^2}}}{2}} \right]_1^2} dy \cr
& = \int_{ - 2}^2 {\left[ {\frac{{{2^2}{y^2}}}{2} - \frac{{{1^2}{y^2}}}{2}} \right]} dy \cr
& = \int_{ - 2}^2 {\frac{3}{2}{y^2}} dy \cr
& {\text{Integrate with respect to }}y \cr
& = \frac{1}{2}\left( {\frac{{{y^3}}}{6}} \right)_{ - 2}^2 \cr
& = \frac{1}{2}\left( {\frac{8}{6} + \frac{8}{6}} \right) \cr
& = 8 \cr} $$
