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QUESTION


\begin{description}

\item[(a)]
Sketch the regiondefined by the inequalities:

$$0\leq x\leq\pi,\ 0\leq y\leq2\pi,\ 0\leq z\leq\frac{\pi}{2}.$$

\item[(b)]
If the region is occupied by a solid $S$ with density at any point
$(x,y,z)$ given by the formula $3x^2y\sin\ z$, compute the total
mass of the region $S$ by eveluating an appropriate triple
integral.

\item[(c)]
The region $S$ is divided by the plane $y=ax$ (where $a$ is a
constant $0<a<2$) into two regions: the region $S_1$ contains the
point $(\pi,0,0)$ and the region $S_2$ contains the point
$(0,2\pi,0)$. Sketch the two regions $S_1$ and $S_2$, and find the
mass of $S_1$ in terms of $a$.

\item[(d)]
Using your answers to parts (b) and (c), find the mass of the
upper part $s_2$, again in terms of $a$, and find the value of $a$
for which the two regions have equal mass.

\end{description}




ANSWER


\begin{description}

\item[(a)]

\setlength{\unitlength}{1cm}

\begin{picture}(9,7)

\put(1,1){\line(4,1){7}}\put(7.5,2.5){$y=ax$}

\put(1,1){\vector(1,0){6.5}}\put(7.5,.5){$x$}
\put(1,1){\vector(0,1){5}}\put(.8,6.2){$z$}
\put(.7,.7){0}\put(.5,1.8){$\frac{\pi}{2}$}\put(2.8,.5){$\pi$}
\put(1,1){\line(2,1){.4}}\put(1.8,1.4){\line(2,1){.4}}\put(2.6,1.8){\line(2,1){.4}}\put(3,2){\line(2,1){4}}
\put(1,2){\line(1,0){2}} \put(3,1){\line(0,1){1}}
\put(1,2){\line(2,1){4}} \put(5,4){\line(1,0){2}}
\put(7,4){\line(0,-1){1}} \put(3,1){\line(2,1){4}}
\put(6.5,4.2){$(\pi,2\pi,\frac{\pi}{2})$} \put(4.5,2.2){S}

\put(5,4){\line(0,-1){.4}}\put(5,3.2){\line(0,-1){.2}}
\put(5,3){\line(1,0){.4}}\put(5.8,3){\line(1,0){.4}}\put(6.6,3){\line(1,0){.4}}
\put(5.2,3.5){$2\pi$}

\end{picture}

\item[(b)]

\begin{eqnarray*}
\int_0^\pi3x^2\,dx\int_0^{2\pi}y\,dy\int_0^{\frac{\pi}{2}}\sin
z\,dz&=&\left[x^3\right]_0^\pi\left[\frac{y^2}{2}\right]_0^{2\pi}\left[-\cos
z\right]_0^{\frac{\pi}{2}}\\ &=&\pi^3\frac{4\pi^2}{2}=2\pi^5
\end{eqnarray*}

\item[(c)]

\setlength{\unitlength}{1cm}

\begin{picture}(12,5)

\put(.5,1.7){$\mathbf{0}$} \put(2.2,.5){$(\pi,0,0)$}
\put(4,1.5){$(\pi,\frac{\pi}{a},0)$} \put(5.8,1.5){$\mathbf{0}$}
\put(7.1,1.5){$(\pi,\frac{\pi}{a},0)$}
\put(11.2,4.2){$(\pi,2\pi,\frac{\pi}{2})$} \put(2.5,2.5){$S_1$}
\put(8.5,3.4){$S_2$}

\put(1,3){\line(1,0){3}}\put(1,3){\line(3,-2){1.5}}
\put(2.5,2){\line(3,2){1.5}} \put(2.5,2){\line(0,-1){1}}
\put(4,3){\line(0,-1){1}} \put(1,2){\line(3,-2){1.5}}
\put(2.5,1){\line(3,2){1.5}} \put(1,2){\line(0,1){1}}

\put(6,2){\line(1,0){2}} \put(6,2){\line(0,1){1}}
\put(8,2){\line(3,1){3}} \put(8,2){\line(0,1){1}}
\put(11,3){\line(0,1){1}} \put(6,3){\line(1,0){2}}
\put(8,3){\line(3,1){3}} \put(9,4){\line(1,0){2}}
\put(6,3){\line(3,1){3}}

\end{picture}

\begin{eqnarray*}
\textrm{Mass}S_1&=&\int_0^\frac{\pi}{2}\!\int_{x=0}^\pi\!\int_{y=0}^{ax}3x^2y\sin
z\,dydxdz\\ &=&\int_0^\frac{\pi}{2}\int_0^\pi\left[3x^2\sin
z\frac{y^2}{2}\right]_0^{ax}\,dxdz\\
&=&\int_0^\frac{\pi}{2}\!\int_0^\pi\frac{3a^2}{2}x^4\sin z\,dxdz\\
&=&\int_0^\frac{\pi}{2}\left[\frac{3a^2}{2}\frac{x^5}{5}\sin
z\right]_0^\pi\,dz\\
&=&\int_0^\frac{\pi}{2}\frac{3}{10}a^2\pi^5\sin z\,dz\\
&=&\left[\frac{3}{10}a^2\pi^5(-\cos z)\right]_0^\frac{\pi}{2}\\
&=&\frac{3}{10}a^2\pi^5
\end{eqnarray*}

\item[(d)]

\begin{eqnarray*}
\textrm{mass}S_2&=&\textrm{mass}S-\textrm{mass}S_1\\
&=&2\pi^5-\frac{3}{10}a^2\pi^5=\left(\frac{20-3a^2}{10}\right)\pi
\end{eqnarray*}

Mass $S_1$=Mass
$S_2\Leftrightarrow\frac{3}{10}a^2=\frac{20-3a^2}{10}\Leftrightarrow6
a^2=20\Leftrightarrow a^2=\frac{10}{3}\Leftrightarrow
a=\sqrt{\frac{10}{3}}$

\end{description}




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