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{\bf Question}

Sketch the region bounded by $z=x^2+y^2$, $z=0$, $x=-a$, $x=a$,
$y=-a$, $y=a$, (where $a$ is some unspecified constant) and
calculate its volume by evaluating a suitable double integral.


{\bf Answer}

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Volume of region = $\displaystyle \int_{y=-a}^{y=a}
\left\{\int_{x=-a}^{x=a} (x^2+y^2) \,dx \right\}\,dy$

\begin{eqnarray*} & = & \int_{-a}^a \left[\frac{x^3}{3}+y^2x
\right]_{x=-a}^{x=a} \,dy = 2\int_{-a}^a \frac{a^3}{3}+ay^2 \,dy\\
& = & 2\left[\frac{a^3y}{3}+\frac{ay^3}{3}\right]_{-a}^a =
\frac{8a^4}{3} \end{eqnarray*}



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