\documentclass[12pt]{article}
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\begin{document}
\begin{large}
\noindent
\makebox[\textwidth]{
	\textbf{Quiz 6 | MA261 | July 14, 2017}
}
\makebox[\textwidth]{Christina Jamroz, Alden Bradford}
\begin{enumerate}
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%%% PROBLEMS

\item Find the volume of the region bounded by the paraboloid $z=4-x^2-y^2$ and the $xy$ co\"ordinate plane by changing to cylindrical co\"ordinates.

\key{
\begin{align*}
\int_0^{2\pi}\int_0^2\int_0^{4-r^2}r\;dzdrd\theta&=\int_0^{2\pi}d\theta\int_0^2(4-r^2)r\;dr\\
&=2\pi\qty(2r^2-\frac{r^4}{4})\Big|_0^2\\
&=8\pi
\end{align*}
}

\item Evaluate $\displaystyle\iiint_R (x^2+y^2) \;dV$, where $R$ is the region between the spheres of radius 1 and 2 centered at the origin.

\emph{Hint:} $\displaystyle\int_0^\pi \sin^3 u \;du=4/3$

\key{\begin{align*}
\iiint_R (x^2+y^2) \;dV&=\int_0^{2\pi}\int_0^\pi\int_1^2(\rho^2\sin^2\phi)\rho^2\sin\phi\; d\rho d\phi d\theta\\
&=\int_0^{2\pi}d\theta\int_0^\pi\sin^3\phi\;d\phi\int_1^2\rho^4\;d\rho\\
&=(2\pi)(4/3)(31/5)\\
&=248\pi/15
\end{align*}
}

\end{enumerate}
\end{large}
\end{document}

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