\documentclass[12pt]{amsart}

% Package Being Used:

\usepackage{enumitem}
\usepackage{bigints}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{graphicx}
\usepackage{psfrag}
\usepackage{color}
\usepackage{url}
\usepackage{algpseudocode}
\usepackage{fancyhdr}
\usepackage{xy}
\input xy
\xyoption{all}

%\usepackage{soul} % this is a package used only temporary that allows us to underline comments that we are planning to remove in future versions of the paper.

% Paper Format and Geometry:

\voffset=-1.4mm
\oddsidemargin=17pt
\evensidemargin=17pt
\topmargin=26pt
\headheight=9pt     
\textheight=576pt
\textwidth=440.8pt
\parskip=0pt plus 4pt

% Head Labels:

\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
%\fancyhead[CO]{\fontsize{9}{11}\selectfont ATOMICITY AND BOUNDEDNESS OF MONOTONE PUISEUX MONOIDS}
%\fancyhead[LE,RO]{\thepage}

% Theorems-like Format and Numbering:


\newtheorem{theorem}{Theorem}%[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}

% Personalized Commands:

\newcommand\cc{\mathbb{C}}
\newcommand\ff{\mathbb{F}}
\newcommand\nn{\mathbb{N}}
\newcommand\qq{\mathbb{Q}}
\newcommand\rr{\mathbb{R}}
\newcommand\zz{\mathbb{Z}}

\newcommand\pval{\mathsf{v}_p}

\newcommand{\twopf}[4]
{
	\left\{
	\begin{array}{ll}
		#1 & \mbox{if } #2 \\
		#3 & \mbox{if } #4
	\end{array}
	\right.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
	
%	\mbox{}
%	\title{Vector Calculus (MATH 53)}
%	
%	\maketitle
	
	
\section*{Triple Integrals}
\bigskip

\begin{problem}[Stewart, Exercise~15.6.29]
	Express the integral $\iiint_E f(x,y,z) \, dV$ as an iterated integral in six different ways, where $E$ is the solid bounded by $y = 4 - x^2 - 4z^2$ and $y = 0$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.6.33]
	Sketch the solid whose volume is given by the triple integral
	\[
		\int_0^1 \int_{\sqrt{x}}^1 \int_0^{1-y} f(x,y,z) \, dz dy dx.
	\]
	Write five other iterated integrals that are equal to the given iterated integral. [Note: Sketch given in the book.]
\end{problem}

\begin{problem}[Stewart, Exercise~15.6.36]
	Write five other iterated integrals that are equal to the iterated integral
	\[
		\int_0^1 \int_y^1 \int_0^z f(x,y,z) \, dx dz dy.
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.6.(37,38)]
	Use geometry to evaluate the following double integrals.
	\begin{enumerate}
		\item $\iiint_C (4 + 5 x^2yz^2) \, dV$, where $C$ is the cylindrical region given by $x^2 + y^2 \le 4$ and $-2 \le z \le 2$.
		\vspace{5pt}
		\item $\iiint_B (z^3 + \sin y + 3) \, dV$, where $B$ is the unit ball $x^2 + y^2 + z^2 \le 1$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercise~15.6.42]
	Find the mass and center of mass of the tetrahedron in the first octant bounded by the plane $x+y+z = 1$ with density given by $\rho(x,y,z) = y$.
\end{problem}

	
\vspace{20pt}

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