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%	\mbox{}
%	\title{Vector Calculus (MATH 53)}
%	
%	\maketitle
	

\section*{Polar and Cylindrical Coordinates}
\bigskip

\begin{problem}[Stewart, Exercise~15.3.(15,17)]
	Use double integrals to find the areas of the following regions.
	\begin{enumerate}
		\item One loop of the rose $r = \cos 3 \theta$.
		\vspace{5pt}
		\item The region inside the circle $(x-1)^2 + y^2 = 1$ and outside the circle $x^2 + y^2 = 1$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercise~15.3.27]
	Use polar coordinate to find the volume of the solid inside both the cylinders $x^2 + y^2 = 4$ and the ellipsoid $4x^2 + 4y^2 + z^2 = 64$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.3.(29,30)]
	Evaluate the following integrals:
	\[
		(1) \ \int_0^2 \int_0^{\sqrt{4 - x^2}} e^{-x^2 - y^2} \, dy dx \quad \quad \quad (2) \ \int_0^a \int_{-\sqrt{a^2 - y^2}}^{\sqrt{a^2 - y^2}} (2x + y) \, dx dy.
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.3.35]
	A swimming pool is circular with a $40$-feet diameter. The depth is constant along east-west lines and increases linearly from $2$ feet at the south end to $7$ feet at the north end. Find the volume of water in the pool.
\end{problem}

\begin{problem}[Stewart, Exercise~15.7.(15,16)]
	In each case, sketch the solid whose volume is given by the following integrals:
	\begin{enumerate}
		\item $\int_{-\pi/2}^{\pi/2} \int_0^2 \int_0^{r^2} r \, dz dr d\theta$,
		\vspace{5pt}
		\item $\int_0^2 \int_0^{2\pi} \int_0^r r \, dz d\theta dr$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercise~15.7.21]
	Evaluate $\iiint_E x^2 \, dV$, where $E$ is the solid that lies within the cylinder $x^2 + y^2 = 1$, above the plane $z = 0$, and below the cone $z^2 = 4x^2 + 4y^2$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.7.25]
	Find the volume of the region $E$ that lies between the paraboloid $z = 24 - x^2 - y^2$ and the cone $z = 2 \sqrt{x^2 + y^2}$. Find the centroid of $E$ (i.e., the center of mass in constant density).
\end{problem}

\begin{problem}[Stewart, Exercise~15.7.29]
	Evaluate
	\[
		\int_{-2}^2 \int_{-\sqrt{4 - y^2}}^{\sqrt{4 - y^2}} \int_{\sqrt{x^2 + y^2}}^2 xz \, dz dx dy.
	\]
	[Hint: Use cylindrical coordinates.]
\end{problem}


\vspace{20pt}

\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}