Question

Find the mass of the solid bounded by z = 1 and , 2 2 z = x + y the density function

being d (x, y, z) = | x | .

Answer 1

Let $\Omega$ be the region bounded by z = 1 and z = x² + y² .

Then $\int\limits_{\Omega}|x|dx dy dz=\int\limits_0^1\int\limits_{-\sqrt{z}}^{\sqrt{z}}\int\limits_{-\sqrt{z-y^2}}^{\sqrt{z-y^2}}|x|dxdydz=\int\limits_0^1\int\limits_{-\sqrt{z}}^{\sqrt{z}}\int\limits_{0}^{\sqrt{z-y^2}}2|x|dxdydz=\int\limits_0^1\int\limits_{-\sqrt{z}}^{\sqrt{z}}x^2|_0^{\sqrt{z-y^2}}dydz=\int\limits_0^1\int\limits_{-\sqrt{z}}^{\sqrt{z}}(z-y^2)dydz=\int\limits_0^1(zy-y^3/3)|_{-\sqrt{z}}^{\sqrt{z}}dydz=\int\limits_0^1\frac{4}{3}z^{2/3}dz=\frac{4}{3}\frac{3}{5}z^{5/3}|_0^1=4/5$

Answer. The mass of the solid equals to 4/5