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Question 5.6.1: Evaluate ∭π xy cos yz dV, where π is the parallelepiped {(x,...

Evaluate ∭πxycos⁡yz dV\iiint_{\pi} x y \cos y z  d V, where π\pi is the parallelepiped {(x,y,z):0≤x≤1,0≤y≤1,0≤z≤π2}\left\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq \frac{\pi}{2}\right\}.

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∭πxycos⁡yz dV=∫01∫01∫0π/2xycos⁡yz dz dy dx=∫01∫01{xy⋅1ysin⁡yz∣0π/2}dy dx=∫01∫01xsin⁡(π2y)dy dx=∫01{−2πxcos⁡π2y∣01}dx=∫012πx dx=x2π∣01=1π.\begin{aligned}\iiint_{\pi} x y \cos y z  d V &=\int_{0}^{1} \int_{0}^{1} \int_{0}^{\pi / 2} x y \cos y z  d z  d y  d x \\&=\int_{0}^{1} \int_{0}^{1}\left\{\left.x y \cdot \frac{1}{y} \sin y z\right|_{0} ^{\pi / 2}\right\} d y  d x=\int_{0}^{1} \int_{0}^{1} x \sin \left(\frac{\pi}{2} y\right) d y  d x \\&=\int_{0}^{1}\left\{-\left.\frac{2}{\pi} x \cos \frac{\pi}{2} y\right|_{0} ^{1}\right\} d x=\int_{0}^{1} \frac{2}{\pi} x  d x=\left.\frac{x^{2}}{\pi}\right|_{0} ^{1}=\frac{1}{\pi} .\end{aligned}

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