Answer
$$4$$
Work Step by Step
$$\eqalign{
& \int_0^\pi {\int_0^\pi {\int_0^{\sin x} {\sin y{\text{ }}} dzdxdy} } \cr
& {\text{Integrate with respect to }}z \cr
& = \int_0^\pi {\int_0^\pi {\left[ {z\sin y} \right]_0^{\sin x}dxdy} } \cr
& = \int_0^\pi {\int_0^\pi {\left[ {\sin x\sin y - 0\sin y} \right]dxdy} } \cr
& = \int_0^\pi {\int_0^\pi {\sin x\sin ydxdy} } \cr
& {\text{Integrate with respect to }}x \cr
& = - \int_0^\pi {\left[ {\sin y\cos x} \right]_0^\pi dy} \cr
& = - \int_0^\pi {\left[ {\sin y\cos \pi - \sin y\cos 0} \right]dy} \cr
& = - \int_0^\pi {\left( { - \sin y - \sin y} \right)dy} \cr
& = 2\int_0^\pi {\sin ydy} \cr
& {\text{Integrate }} \cr
& = 2\left[ { - \cos y} \right]_0^\pi \cr
& = - 2\left[ {\cos \pi - \cos 0} \right] \cr
& = - 2\left( { - 1 - 1} \right) \cr
& = 4 \cr} $$
