\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

QUESTION


Sketch the region defined by the inequalities $0\leq x\leq\pi,
0\leq y\leq\pi, 0\leq z\leq\pi$.

\begin{description}

\item[(i)]
If the region is occupied by a solid $S$ whose density at the
point $(x,y,z)$ is $3y^2z\sin(x)$, calculate the total mass of $S$
using an appropriate triple integral.

\item[(ii)]
The plane $z=ay$ (where $0<a\leq1$ is a constant) divides $S$ into
two parts: a lower part $S_1$ lying below the plane, and an upper
part $S_2$ lying above the plane. Sketch $S_1$ and $S_2$ when
$a=1$ and when $0<a<1$. Find the mass of the lower part $S_1$ in
terms of $a$.

\item[(iii)]
Using your answers to (i) and (ii), find the mass of the upper
part $S_2$ in terms of $A$. Hence show that the mass of $S_1$ is
equal to the mass of $S_2$ when

$$a=\sqrt{\frac{5}{6}}.$$

\end{description}




ANSWER


\setlength{\unitlength}{.5cm}

\begin{picture}(12,12)

\put(5,5){\vector(1,0){6}} \put(5,5){\vector(0,1){6}}
\put(5,5){\vector(-1,-1){4}} \put(5.2,11){$z$} \put(11.2,5){$y$}
\put(.5,1){$x$}

\put(2,1){$\pi$}\put(9,4.3){$\pi$}\put(4.2,9){$\pi$}

\put(5,9){\line(1,0){4}}\put(9,9){\line(-1,-1){3}}
\put(6,6){\line(-1,0){4}}\put(2,6){\line(1,1){3}}
\put(9,9){\line(0,-1){4}} \put(6,6){\line(0,-1){4}}
\put(2,6){\line(0,-1){4}} \put(2,2){\line(1,0){4}}
\put(6,2){\line(1,1){3}}

\end{picture}

\begin{description}

\item[(i)]
$S$ is a solid cube with side length $\pi$. Mass of $S$:

\begin{eqnarray*}
&&3\int_0^\pi z\,dz\int_0^\pi y^2\,dy\int_0^\pi\sin(x)\,dx\\
&=&3\left[\frac{1}{2}z^2\right]_0^\pi\left[\frac{1}{3}y^3\right]_0^\pi
\left[-\cos(x)\right]_0^\pi\\
&=&3\left(\frac{1}{2}\pi^2\right)\left(\frac{1}{3}\pi^3\right)
\left(-\cos(\pi)+\cos(0)\right)\\ &=&\pi^5
\end{eqnarray*}

\item[(ii)]

when $a=1,\ z=y$


\begin{picture}(12,12)

\put(5,5){\vector(1,0){6}} \put(5,5){\vector(0,1){6}}
\put(5,5){\vector(-2,-3){3}} \put(5.2,11){$z$} \put(11.2,5){$y$}
\put(1,1){$x$}

\put(5,9){\line(1,0){4}}\put(9,9){\line(-2,-3){2}}
\put(7,6){\line(-1,0){4}}\put(3,6){\line(2,3){2}}
\put(9,9){\line(0,-1){4}} \put(7,6){\line(0,-1){4}}
\put(3,6){\line(0,-1){4}} \put(3,2){\line(1,0){4}}
\put(7,2){\line(2,3){2}}

\put(3,2){\line(1,1){4}} \put(5,5){\line(1,1){4}}

\put(6,7){$S_2$} \put(7,4){$S_1$}

\end{picture}

when $0<a\leq1,\ z=ay$

\begin{picture}(12,12)

\put(5,5){\vector(1,0){6}} \put(5,5){\vector(0,1){6}}
\put(5,5){\vector(-1,-1){4}} \put(5.2,11){$z$} \put(11.2,5){$y$}
\put(.5,1){$x$}

\put(5,9){\line(1,0){4}}\put(9,9){\line(-1,-1){3}}
\put(6,6){\line(-1,0){4}}\put(2,6){\line(1,1){3}}
\put(9,9){\line(0,-1){4}} \put(6,6){\line(0,-1){4}}
\put(2,6){\line(0,-1){4}} \put(2,2){\line(1,0){4}}
\put(6,2){\line(1,1){3}}

\put(5,5){\line(2,1){4}} \put(2,2){\line(2,1){4}}
\put(6,4){\line(1,1){3}}

\put(5,7){$S_2$} \put(7.5,4){$S_1$}

\end{picture}

The plane is \lq\lq hinged'' at the $x$-axis.

\begin{eqnarray*}
\textrm{Mass of }S_1&=&
\int_{y=0}^{y=\pi}\!\int_{z=0}^{z=ay}\!\int_{x=0}^{x=\pi}3y^2z\sin(x)\,dxdydz\\
&=&\int_{y=0}^{y=\pi}\!\int_{z=o}^{z=ay}6y^2z\,dzdy\\
&=&\int_{y=0}^{y=\pi}\left[3y^2z^2\right]_{z=0}^{z=ay}\,dy\\
&=&\int_{y=0}^{y=\pi}3a^2y^4\,dy\\
&=&\left[\frac{3}{5}a^2y^5\right]_0^\pi\\ &=&\frac{3a^2\pi^5}{5}
\end{eqnarray*}

\item[(iii)]

\begin{eqnarray*}
\textrm{Mass of }S_2&=&\textrm{Mass of $s-$ mass of }S_1\\
&=&\pi^5-\frac{3a^2\pi^5}{5}
\end{eqnarray*}

\begin{eqnarray*}
\textrm{mass of $S_1=$ mass of
}S_2&\Rightarrow&\frac{3a^2\pi^5}{5}=\pi^5-\frac{3a^2}{5}\\
&\Rightarrow&\frac{6a^2\pi^5}{5}=\pi^5\\
&\Rightarrow&a^2=\frac{5}{6}\\ &\Rightarrow&a=\sqrt{\frac{5}{6}}
\end{eqnarray*}

We take the positive square root since $0<a\leq1$.

\end{description}





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