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\flushleft\underline{Homework 8 --- MATH 1586 Spring 2018} \\

\begin{enumerate}[1.]
\item Draw the region and calculate $\displaystyle\int_2^3 \displaystyle\int_0^6 x^2y \mathrm{d}x\mathrm{d}y$.
\item Draw the region and calculate $\displaystyle\int_0^1 \displaystyle\int_1^2 \dfrac{e^x}{y} \mathrm{d}y\mathrm{d}x$. 
\item Sketch $R$ and calculate $\displaystyle\iint\limits_R x^3 y^2 \mathrm{d}A$, where $R$ is the region bounded by the lines $y=2x$, $y=0$, and $x=2$.
\item Sketch $R$ and calculate $\displaystyle\iint\limits_R e^{xy} \mathrm{d}A$ where $R$ is the region bounded by the lines $y=2-2x$, $x=0$, and $y=0$.
\item Sketch $R$ and calculate $\displaystyle\iint\limits_R x+y \mathrm{d}A$, where $R$ is the region bounded by $y=0$, $y=x^2$, and $x=1$.
\item Sketch $R$ and calculate $\displaystyle\iint\limits_R x^2+y \mathrm{d}A$, where $R$ is bounded between $y=\sqrt{x}$ and $y=x^2$.
\item Sketch $R$ and calculate $\displaystyle\iint\limits_R e^xy \mathrm{d}A$, where $R$ is the triangular region defined by the points $(-1,0), (1,0)$, and $(0,1)$. \\
(\textit{hint: recall that the line between the points $(a,b)$ and $(c,d)$ has slope $m=\dfrac{b-d}{c-a}$ and obeys the equation $y-b=m(x-a)$})
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