Example 3.12.1.

A solid \(E\) lies within the cylinder \(x^2 + y^2 = 1\), below the plane \(z = 4\), and above the paraboloid \(z = 1 -x^2 -y^2\). The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of \(E\).

\[E = \big \{\big (r,\theta ,z\big )\hspace {0.1cm}\big |\hspace {0.1cm} 0\leq r \leq 1\hspace {0.1cm} 0 \leq \theta \leq 2\pi \hspace {0.1cm}, \hspace {0.1cm} 1 - r^2 \leq z \leq 4\big \}\]

Density function \(\hspace {0.2cm}f(x,y,z) = K\sqrt {x^2 + y^2} = K r\)

\begin {align*} M & = \iiint \limits _EK\sqrt {x^2 + y^2}\hspace {0.2cm} dV\\\\ & = \int ^{2\pi }_0 \int ^1_0 \int ^4_{1-r^2} K r\hspace {0.1cm}rdz\hspace {0.1cm} dr \hspace {0.1cm} d\theta \\\\ & = \int ^{2\pi }_0 d\theta \int ^1_0 \int ^4_{1-r^2}K\,r^2\,dz\, dr\\\\ & = \frac {12\pi K}{5}\\ \end {align*}

Example 3.12.2.

Evaluate \(\displaystyle {\int ^2_{-2}\int ^{\sqrt {4-x^2}}_{-\sqrt {4 - x^2}}\int ^2_{\sqrt {x^2 + y^2}}(x^2 + y^2)\, dz \,dy \,dx}\)

\[E= \big \{ \big (r,\theta ,z\big )\hspace {0.1cm}\big | \hspace {0.1cm} 0\leq r \leq 2 \hspace {0.1cm}, \hspace {0.1cm} 0 \leq \theta \leq 2\pi \hspace {0.1cm},\hspace {0.1cm} r \leq z \leq 2\big \}\]

\begin {align*} \int ^2_{-2}\int ^{\sqrt {4-x^2}}_{-\sqrt {4 - x^2}}\int ^2_{\sqrt {x^2 + y^2}}(x^2 + y^2) dz dy dx & = \int ^2_0\int ^{2\pi }_0\int ^2_r r^2\hspace {0.1cm} r\,dz\, dr\,d\theta \\\\ & = \int ^2_0d\theta \int ^{2\pi }_0\int ^2_r r^3\, dz \,dr\\\\ & = \frac {16\pi }{5}\\\\ \end {align*}