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


\section*{Spherical Coordinates}
\bigskip

\begin{problem}[Stewart, Exercise~15.8.(17,18)]
	In each case, sketch the solid whose volume is given by the following integrals:
	\begin{enumerate}
		\item $\int_0^{\pi/6} \int_0^{\pi/2} \int_0^3 \rho^2 \sin \phi \, d\rho d\theta d\phi$,
		\vspace{5pt}
		\item $\int_0^{\pi/4} \int_0^{2\pi} \int_0^{\sec \phi} \rho^2 \sin \phi \, d\rho d\theta d\phi$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercise~15.8.27]
	Find the volume of the part of the ball $\rho \le a$ that lies between the cones $\phi = \pi/6$ and $\phi = \pi/3$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.8.28]
	Find the average distance from a point inside a ball of radius $a$ to its center.
\end{problem}

\begin{problem}[Stewart, Exercise~15.8.30]
	Find the volume of the solid that lies within the sphere $x^2 + y^2 + z^2 = 4$, above the $xy$-plane, and below the cone $z = \sqrt{x^2 + y^2}$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.8.41]
	Evaluate
	\[
		\int_0^1 \int_0^{\sqrt{1 - x^2}} \int_{\sqrt{x^2 + y^2}}^{\sqrt{2 - x^2 - y^2}} xy \, dz dy dx.
	\]
	[Hint: Use spherical coordinates.]
\end{problem}

\begin{problem} [Cal Final]
	Let $E$ be the region defined by the inequalities
	\[
		x^2 + y^2 + z^2 \le 4, \quad 0 \le y \le x, \quad z \ge 0.
	\]
	Calculate the total mass of $E$ if the mass density is given by $z^2$.
\end{problem}

	
\vspace{20pt}

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