The density of the solid of Example 2 is given by \rho(x, y, z)=x+2 y+4 z \mathrm{~kg} / \mathrm{m}^{3}. Calculate the total mass of the solid.
Question 5.6.5: The density of the solid of Example 2 is given by p(x, y, z)...
\begin{aligned}\mu(S) &=\int_{0}^{1} \int_{x^{2}}^{x} \int_{x-y}^{x+y}(x+2 y+4 z) d z d y d x=\int_{0}^{1} \int_{x^{2}}^{x}\left\{\left.\left[(x+2 y) z+2 z^{2}\right]\right|_{x-y} ^{x+y}\right\} d y d x \\&\left.=\int_{0}^{1} \int_{x^{2}}^{x}\left(10 x y+4 y^{2}\right) d y d x=\left.\int_{0}^{1}\left\{5 x y^{2}+\frac{4 y^{3}}{3}\right)\right|_{x^{2}} ^{x}\right\} d x \\&=\int_{0}^{1}\left(5 x^{3}-5 x^{5}+\frac{4}{3} x^{3}-\frac{4}{3} x^{6}\right) d x=\int_{0}^{1}\left(\frac{19}{3} x^{3}-5 x^{5}-\frac{4}{3} x^{6}\right) d x \\&=\frac{19}{12}-\frac{5}{6}-\frac{4}{21}=\frac{47}{84} \mathrm{~kg} .\end{aligned}
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