Evaluate the following integral.
\[
\int_{0}^{\pi} \int_{0}^{1} \int_{0}^{\pi / 2} \sin \pi x \cos y \sin z d y d x d z
\]
\[
\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi / 2} \sin \pi x \cos y \sin z d y d x d z=\square
\]
(Type an exact answer, using $\pi$ as needed.)

Answer

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Answer

The final answer to the problem is $\boxed{-\frac{\sin(1)\cos(\pi^2)}{\pi} + \frac{\sin(1)}{\pi}}$

Steps

Step 1 ：Define the function $f = \sin(z) \sin(\pi x) \cos(y)$

Step 2 ：Integrate $f$ with respect to $y$ from 0 to 1, resulting in $\sin(1) \sin(z) \sin(\pi x)$

Step 3 ：Integrate the result with respect to $x$ from 0 to $\pi$, resulting in $-\sin(1) \sin(z) \cos(\pi^2)/\pi + \sin(1) \sin(z)/\pi$

Step 4 ：Finally, integrate the result with respect to $z$ from 0 to $\pi/2$, resulting in $-\sin(1) \cos(\pi^2)/\pi + \sin(1)/\pi$

Step 5 ：The final answer to the problem is $\boxed{-\frac{\sin(1)\cos(\pi^2)}{\pi} + \frac{\sin(1)}{\pi}}$