Triple Integral with Trigonometric Function

Evaluate the triple integral from 0 to $\pi$ and 0 to 2 and then 0 to $\sqrt{4 - Z^2}$ of $Z \sin Y$ with respect to $dX$ $dZ$ $dy$.

This problem involves the evaluation of a triple integral which requires understanding integral calculus in three dimensions. When evaluating such integrals, one must decide the order of integration and consider any variable dependencies. In this problem, the integrals are set with specific bounds for Z, Y, and X, which gives a clear sequence for integration, typically working from the innermost integral outward. This approach helps manage complexity, as each integration potentially simplifies the problem for the next variable.

Additionally, this integral incorporates a trigonometric function, sine of Y, which demonstrates how such functions behave under integration over a specified volume. The limits of integration involve square roots and simple constants, indicating a bounded geometric region of integration which might be spherical or cylindrical depending on the arrangement of limits. This aspect of the problem can lead to exploring whether changing the order of integration or using symmetry might simplify the process, a common strategy in multivariable calculus when facing complex integrals.