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QUESTION

\begin{description}

\item[(a)]
Sketch the region defined by the inequalities $x^2+y^2\leq x,
0\leq z\leq3$. If the region is occupied by a solid whose density
at the point $(x,y,z)$ is $x^2+y^2+z$ calculate its total mass by
means of an appropriate triple integral.

\item[(b)]
A cube having side length 2 has density at a point given by twice
the square of its distance from the center of the cube. Find the
mass of the cube.

\end{description}



ANSWER


\begin{description}

\item[(a)]

DIAGRAM

\begin{eqnarray*}
\textrm{Mass}&=&\int\!\!\!\int\!\!\!\int_Rx^2+y^2+z\,dV\\
&=&\int\!\!\!\int\!\!\!\int_R(\rho^2+z)\rho\,dzd\rho d\phi
\end{eqnarray*}

in cylindrical coordinates.

$$R=\{(\rho,\phi,z)|0\leq\rho\leq\sqrt{3}, 0\leq\phi\leq2\pi,
\rho^2\leq z\leq3\}$$

so

\begin{eqnarray*}
\textrm{Mass}&=&\int_{\phi=0}^{2\pi}\!\int_{\rho=0}^{
\sqrt{3}}\!\int_{z=\rho^2}^3(\rho^3+\rho z)\,dzd\rho d\phi\\
&=&\int_0^{2\pi}\!\int_0^{\sqrt{3}}\left[\rho^3
z+\rho\frac{z^2}{2}\right]_{\rho^2}^3\,dsd\phi\\
&=&\int_0^{2\pi}\!\int_0^{\sqrt{3}}\left(3\rho^3+\frac{9}{2}
\rho-3\frac{\rho^5}{2}\right)\,d\rho d\phi\\
&=&\int_0^{2\pi}\left[\frac{3}{4}\rho^4+\frac{9\rho^2}{4}-
\frac{\rho^6}{4}\right]_0^{\sqrt{3}}\,d\phi\\
&=&\int_0^{2\pi}\left(\frac{27}{4}+\frac{27}{4}-
\frac{27}{4}\right)\,d\phi\\ &=&\frac{27}{2}\pi
\end{eqnarray*}

\item[(b)]
The cube consists of eight identical octants so its mass is given
by

\begin{eqnarray*}
&&8\int_0^1\!\int_0^1\!\int_0^12(x^2+y^2+z^2)\,dxdydz\\
&=&8\int_0^1\!\int_0^1\left[\frac{2x^3}{3}+2(y^2+z^2)x\right]_0^1\,dydz\\
&=&8\int_0^1\!\int_0^1\left(\frac{2}{3}+2y^2+2z^2\right)\,dydz\\
&=&8\int_0^1\left[\left(\frac{2}{3}+2z^2\right)y+2\frac{y^3}{3}\right]_0^1\,dz\\
&=&8\int_0^1\left(\frac{2}{3}+2z^2+\frac{2}{3}\right)\,dz\\
&=&8\left[\frac{4z}{3}+2\frac{z^3}{3}\right]_0^1\\ &=&16
\end{eqnarray*}

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