Answer
0
Work Step by Step
$$\eqalign{
& \int_{ - 1}^1 {\int_1^2 {\int_0^1 {6xyz\,dydxdz} } } \cr
& {\text{Integrate with respect to }}y \cr
& = \int_{ - 1}^1 {\int_1^2 {\left[ {3x{y^2}z} \right]_0^1dx} } dz \cr
& = \int_{ - 1}^1 {\int_1^2 {\left( {3xz} \right)dx} } dz \cr
& = 3\int_{ - 1}^1 {z\int_1^2 {xdx} } dz \cr
& {\text{Integrate with respect to }}x \cr
& = 3\int_{ - 1}^1 {z\left[ {\frac{{{x^2}}}{2}} \right]_1^2} dz \cr
& = 3\int_{ - 1}^1 {z\left( {\frac{3}{2}} \right)} dz \cr
& = \frac{9}{2}\int_{ - 1}^1 {zdz} \cr
& {\text{Integrate with respect to }}z \cr
& = \frac{9}{2}\left( {\frac{{{z^2}}}{2}} \right)_{ - 1}^1 \cr
& = \frac{9}{2}\left( {\frac{1}{2} - \frac{1}{2}} \right) \cr
& = 0 \cr} $$
