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

\section*{General Changes of Coordinates}
\bigskip

\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}[UC Berkeley Final]
	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}