\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{Introduction} \label{sec:intro}
	%\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
\section{Double and Triple Integrals}

\begin{problem}[Stewart, Exercise~15.1.11]
	Evaluate the double integral
	\[
		\iint_R (4-2y) \, dA, \quad R = [0,2] \times [0,1],
	\]
	by first identifying it as the volume of a solid.
\end{problem}

\begin{problem}[Stewart, Exercise~15.1.34]
	Evaluate the double integral
	\[
		\iint_R \frac 1{1 + x + y} \, dA, \quad R = [1,3] \times [1,2].
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.1.42]
	Find the volume of the solid in the first octant bounded by the cylinder $z = 16 - x^2$ and the plane $y=5$.
\end{problem}

%\begin{problem}[Stewart, Exercise~15.1.43]
%	Find the volume of the solid enclosed by the paraboloid $z = 2 + x^2 + (y-2)^2$ and the planes $z = 1$, $x = 1$, $x = -1$, $y = 0$, and $y = 4$.
%\end{problem}

\begin{problem}[Stewart, Exercise~15.1.49]
	Use symmetry to evaluate the double integral
	\[
		\iint_R \frac {xy}{1 + x^4} \, dA, \quad R = [-1,1] \times [0,1].
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.22]
	Evaluate the double integral
	\[
		\iint_D y \, dA,
	\]
	where $D$ is the triangular region with vertices $(0,0)$, $(1,1)$, and $(4,0)$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.27]
	Find the volume of the tetrahedron enclosed by the coordinate planes and the plane $2x + y + z = 4$.
\end{problem}

\begin{problem}[Stewart, Example~15.2.4]
	Find the volume of the tetrahedron bounded by the planes $x + 2y + z = 2$, $x=2y$, and $z=0$ using:
	\begin{enumerate}
		\item double integrals,
		\item triple integrals.
	\end{enumerate}
	[Hint: The tetrahedron lies in the first octant. Sketch it by looking from your left.]
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.40]
	Sketch the solid whose volume is given by the integral
	\[
		\int_0^1 \int_0^{1-x^2} (1-x) \, dy dx.
	\]
\end{problem}
\newpage

\begin{problem}[Stewart, Exercise~15.2.(51,52,55)]
	Evaluate the following integrals:
	\begin{enumerate}
		\item $\int_0^1 \int_{3y}^3 e^{x^2} \, dx dy$,
		\vspace{4pt}
		\item $\int_0^1 \int_{x^2}^1 \sqrt{y} \sin y \, dy dx$,
		\vspace{4pt}
		\item $\int_0^1 \int_{\arcsin y}^{\pi/2} \cos x \sqrt{1 + \cos^2x} \, dx dy$.
	\end{enumerate}
	[Hint: Reverse the order of integration first.]
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Calculate the iterated integral
	\[
		\int_0^4 \int_{\sqrt{x}}^2 e^{y^3} \, dydx.
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.(65,67,68)]
	Use geometry to evaluate the following double integrals.
	\begin{enumerate}
		\item $\iint_D (x+2) \, dA$, \quad $D = \{(x,y) \mid 0 \le y \le \sqrt{9 - x^2}\}$,
		\vspace{5pt}
		\item $\iint_D (2x + 3y) \, dA$, \quad $D = [0,a] \times [0,b]$,
		\vspace{5pt}
		\item $\iint_D (2 + x^2y^3 - y^2 \sin x) \, dA$, \quad $D = \{(x,y) \mid |x| + |y| \le 1 \}$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercise~15.4.9]
	Find the mass and center of mass of the lamina that occupies the region bounded by the curves $y=e^{-x}$, $y=0$, $x=0$, and $x=1$ if its density function is $\rho(x,y) = xy$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.5.3]
	Use the formula
	\[
		A(S) = \iint_D \sqrt{1 + f_x^2 + f_y^2} \, dA
	\]
	to find the surface area of the part of the plane $3x + 2y + z = 6$ that lies in the first octant.
\end{problem}

\begin{problem}[Stewart, Exercise~15.5.7]
	Find the surface area of the part of the hyperbolic paraboloid $z = y^2 - x^2$ that lies between the cylinders $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$. [Hint: Use the formula for surface area via double integrals.]
\end{problem}

\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}


\section{Polar, Cylindrical, and Spherical Coordinates}

\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}

\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, Summer 2018W]
	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}

\section{Change of Coordinates}

\begin{problem} [Stewart, Example~15.9.(23,25,27)]
	Evaluate the following integrals.
	\begin{enumerate}
		\item $\iint_R \frac{x - 2y}{3x - y} \, dA$, where $R$ is the parallelogram enclosed by the lines $x - 2y = 0$, $x - 2y = 4$, $3x - y = 1$, and $3x - y = 8$.
		\vspace{5pt}
		\item $\iint_R \cos \big(\frac{y-x}{y+x} \big) \, dA$, where $R$ is the trapezoidal region with vertices $(1,0)$, $(2,0)$, $(0,2)$, and $0,1$.
		\vspace{5pt}
		\item $\iint_R e^{x+y} \, dA$, where $R$ is given by the inequality $|x| + |y| \le 1$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Example~15.9.3]
	 Evaluate $\int \int_R e^{(x+y)/(x-y)} d\,A$, where $R$ is the trapezoidal region with vertices $(1,0)$, $(2,0)$, $(0,-2)$, and $(0,-1)$. [Hint: Define $T$ by $u = x+y$ and $v = x-y$ and find that $T^{-1}$ is defined by $x = 1/2 (u+v)$ and $y = 1/2(u-v)$ and, therefore, $\partial(x,y)/\partial(u,v) = -1/2$][Answer: $3/4(e - e^{-1})$]
\end{problem}

\begin{problem}[Stewart, Exercise~5.9.28]
	Let $f$ be continuous on $[0,1]$ and let $R$ be the triangular region with vertices $(0,0)$, $(1,0)$, and $(0,1)$. Show that
	\[
		\iint_R f(x+y) \, dA = \int_0^1 uf(u) \, du.
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~5.56]
	Find the volume of the region bounded by the surface $\sqrt{x} + \sqrt{y} + \sqrt{z} = 1$ and the coordinate planes. [Hint: Use the change of coordinates $x = u^2$, $y = v^2$, and $z = w^2$.]
\end{problem}

\begin{problem}[Stewart, Exercise~5.57]
	Evaluate $\iint_R xy \, dA$, where $R$ is the square with vertices $(0,0)$, $(1,1)$, $(2,0)$, and $(1,-1)$. [Hint: Use an appropriate change of coordinates.]
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Let $D$ be the region in the plane where $x^2 + 2y^2 \le 1$. Compute the double integral
	\[
		\iint_D (x^2 + 2y^2)^{2018} \, dA.
	\]
	[Hint: Use $u = x$ and $y = \sqrt{2}y$. Answer: $\sqrt{2} \pi/4038$.]
\end{problem}

%\begin{problem} [Stewart, Example~15.9.3]
%	Evaluate $\int \int_R e^{(x+y)/(x-y)} d\,A$, where $R$ is the trapezoidal region with vertices $(1,0)$, $(2,0)$, $(0,-2)$, and $(0,-1)$. [Hint: Define $T$ by $u = x+y$ and $v = x-y$ and find that $T^{-1}$ is defined by $x = 1/2 (u+v)$ and $y = 1/2(u-v)$ and, therefore, $\partial(x,y)/\partial(u,v) = -1/2$][Answer: $3/4(e - e^{-1})$]
%\end{problem}

	
\vspace{20pt}

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