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QUESTION


\begin{description}

\item[(i)]
Sketch the region defined by the inequalities:

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

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

\item[(iii)]
The region $S$ is divided by the plane $x=ay$ (where $a$ is a
constant $0<a<\frac{1}{2}$) 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[(iv)]
Using your answers to parts (i) and (ii), 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[(i)]

\setlength{\unitlength}{1cm}

\begin{picture}(9,7)

\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(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[(ii)]

\begin{eqnarray*}
\textrm{Mass}&=&\int_0^{2\pi}y^2\,dy\int_0^\pi2x\,dx\int_0^{\frac{\pi}{2}}\cos
z\,dz\\
&=&\left[\frac{y^3}{3}\right]_0^{2\pi}\left[x^2\right]_0^\pi\left[\sin
z\right]_0^\frac{\pi}{2}\\ &=&\frac{8\pi^5}{3}
\end{eqnarray*}

\item[(iii)]

\setlength{\unitlength}{1cm}

\begin{picture}(9,7)

\put(1,1){\line(4,1){4}}\put(5.2,1.8){$x=ay$}
\put(1,2){\line(4,1){4}}\put(5,2){\line(0,1){1}}

\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}

\begin{eqnarray*}
\textrm{Mass of }S_1&=&\int_{x=0}^\pi
\!\int_{y=0}^\frac{x}{a}\!\int_{z=0}^\frac{\pi}{2}y^22x\cos
z\,dzdydx\\ &=&\int_{x=0}^\pi\!\int_{y=0}^\frac{x}{a}\left[\sin
z\right]_0^\frac{\pi}{2}y^22x\,dydx\\
&=&\int_{x=0}^\pi\!\int_{y=0}^\frac{x}{a}2xy^2\,dydx\\
&=&\int_{x=0}^\pi\left[2x\frac{y^3}{3}\right]_{y=0}^\frac{x}{a}\,dx\\
&=&\int_0^\pi\frac{2x^4}{3a^3}\,dx\\
&=&\left[\frac{2x^5}{15a^3}\right]_0^\pi=\frac{2\pi^5}{15a^3}
\end{eqnarray*}


\item[(iv)]
mass of $S_2=\left(\frac{8}{3}-\frac{2}{15a^3}\right)\pi^5$

mass $(S_1)=$mass
$(S_2)\Leftrightarrow\frac{4}{15a^3}=\frac{8}{3}\Leftrightarrow
a^3=\frac{1}{10}$ or $a=\frac{1}{\sqrt[3]{10}}$


\end{description}





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