Evaluate \iiint_{S} 2 x^{3} y^{2} z d V, where S is the region \left\{(x, y, z): 0 \leq x \leq 1, x^{2} \leq y \leq x, x-y \leq z \leq x+y\right\}.
Question 5.6.2: Evaluate ∭s 2x^3y^2z dV, where S is the region {(x,y,z):0 &l...
\begin{aligned}\iiint_{S} 2 x^{3} y^{2} z d V &=\int_{0}^{1} \int_{x^{2}}^{x} \int_{x-y}^{x+y} 2 x^{3} y^{2} z d z d y d x \\&=\int_{0}^{1} \int_{x^{2}}^{x}\left\{\left.x^{3} y^{2} z^{2}\right|_{x-y} ^{x+y}\right\} d y d x \\&=\int_{0}^{1} \int_{x^{2}}^{x} x^{3} y^{2}\left[(x+y)^{2}-(x-y)^{2}\right] d y d x \\&=\int_{0}^{1} \int_{x^{2}}^{x} 4 x^{4} y^{3} d y d x=\int_{0}^{1}\left\{\left.x^{4} y^{4}\right|_{x^{2}} ^{x}\right\} d x \\&=\int_{0}^{1}\left(x^{8}-x^{12}\right) d x=\frac{1}{9}-\frac{1}{13}=\frac{4}{117} .\end{aligned}
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