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\author{Alexander J. Tusa}
\title{Change of Variables in 2D}
\begin{document}
	\maketitle
	\begin{enumerate}
		\item Evaluate $\displaystyle\int \int_R\ dA$ where $R$ is the region bounded by the lines $x+y=1$, $x+y=2$, $2x-3y=2$, and $2x-3y=5$. Use the transformation $u=x+y$ and $v=2x-3y$. Draw the region $R$ in the $xy$-plane and the new region $S$ in the $uv$-plane. also, compute $x=x(u,v),\ y=y(u,v)$.

		\item Evaluate $\displaystyle\int\int_R \frac{2x-y}{2}\ dA$ where $R$ is the region bounded by the lines $y=2x$, $y=2x-2$, $y=0$, and $y=4$. Use the transformation $u=2x-y$ and $v=y$. Draw the region $R$ in the $xy$-plane and the new region $S$ in the $uv$-plane. Also, computer $x=x(u,v),$ and $y=y(u,v)$.
		
		\item Evaluate $\displaystyle\int\int_R 2(x-y)\ dA$ where $R$ is the region bounded by the lines $x=0$, $x=-3$, $y=x$ and $y=x+1$. Use the transformation $u=-x$ and $v=-x+y$. Draw the region $R$ in the $xy$-plane and the new region $S$ in the $uv$-plane. Also, compute $x=x(u,v)$, and $y=y(u,v)$.
		
		\item Evaluate $\displaystyle\int\int_R (2x^2-xy-y^2)\ dA$ where $R$ is the region bounded by the lines $y=-2x+4,\ y=-2x+7,\ y=x-2,$ and $y=x+1$ in the first quadrant. Use the transformation $u=x-y$, and $v=2x+y$. Draw the region $R$ in the $xy$-plane and the new region $S$ in the $uv$-plane. Also, compute $x=x(u,v)$ and $y=y(u,v)$.
		
		\item Evaluate $\displaystyle\int\int_R (3x^2+14xy+8y^2)\ dA$ where $R$ is the region bounded by the lines $y=-3/2x+1,\ y=-3/2x+3,\ y=-x/4,$ and $y=-x/4+1$. Use the transformation $u=3x+2y$ and $v=x+4y$. Draw the region $R$ in the $xy$-plane and the new region $S$ in the $uv$-plane. Also, compute $x=x(u,v)$, and $y=y(u,v)$.
	\end{enumerate}
\end{document}