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\begin{center}{ \Large{Worksheets}  \\\large{10/5/2017} }\end{center}
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\item Compute the integral \[\iint\limits_{D}{x(x+y)dA}\] where $D$ is the rectangle defined by $-1\le x\le 1$ and $0\le y\le 1$.
\newpage
\item Fill in the following table\\\\
\begin{tabular}{c|c}
Domain $D$ &  $\iint\limits_{D}{u(x,y)dA}=$ \\
\hline
$a\le x\le b$ and $c\le y\le d$&  \\
\hline
$a\le x\le b$ and $f(x)\le y\le g(x)$ &  \\
\hline
$a\le y\le b$ and $f(y)\le x\le g(y)$ & \\
\hline
\end{tabular}
\vspace{0.3in}
\item For each of the domain $D$ below, express it in form of one of the 3 types in the previous problem.\begin{enumerate}
\item The rectangle with vertices (0,0), (2,0), (2,1), (0,1).\vspace{1.5in}
\item The triangle with vertices (0,0), (2,0), (2,1).\vspace{1.5in}
\item The triangle with vertices (0,0), (2,1), (2,-1).\vspace{1.5in}
\item The triangle with vertices (0,0), (1,2), (2,0).\vspace{1.5in}
\end{enumerate}
\item Compute the integral \[\iint\limits_{D}{x(x+y)dA}\] where $D$ is the triangle with vertices (0,0), (2,0) and (2,1).
\newpage
\item You have learned (or will learn) an important property of integration: Suppose the domain $D$ can be ``split'' into two non-overlapping domains $D_1$ and $D_2$. Then
\[\iint\limits_{D}{u(x,y)dA}=\iint\limits_{{{D}_{1}}}{u(x,y)dA}+\iint\limits_{{{D}_{2}}}{u(x,y)dA}.\]
An application is that: to find the integral of a function over a domain $D$ which is not of any types in Problem 2, we try to split $D$ into subdomains so that each belongs to one of the types in Problem 2. The integration on each subdomain can be computed. Then we add them together.

~~~~~Let's practice! \textbf{Split the following domains into two (or more) subdomains, each of which is of one of the types in Problem 2.}\begin{enumerate}
\item $D$ is the triangle with vertices (1,1), (2,3), (4,2).
\vspace{3.5in}
\item $D$ is the quadrilateral with vertices (0,0), (1,1), (3,2), (4,0).
\end{enumerate}
\end{enumerate}
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