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\begin{center}
  18.022 Recitation Quiz \iftoggle{solutions}{(with solutions)}{} \\
  5 November 2015 
\end{center}

\itm (5.5.30 in \textit{Colley}) Find the volume of the solid that is bounded by the paraboloid $z = 9 - x^2 - y^2$, the $xy$-plane, and the boundary of $[-1,1]\times[-1,1]\times \R$. 

\sol{
  To find the volume of the region, we integrate 1 over the region. We calculate 
\begin{align*}
  \text{volume} &= \int_{-1}^{1}\int_{-1}^{1}\int_{0}^{9-x^2-y^2}1\,dz\,dy\,dx \\
  &= \int_{-1}^{1}\int_{-1}^{1}9-x^2-y^2\,dy\,dx \\
  &= \int_{-1}^{1}[9y-x^2y-y^3/3]_{-1}^{1}\,dx = \boxed{100/3}. \qedhere
\end{align*}
}{\vfill}

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