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\flushleft\underline{Homework 9 --- MATH 1586 Spring 2018} \\
We use polar coordinates to describe circular regions for double integration. In this course, any polar coordinates problem will necessarily have ``$x^2+y^2$" in the function being integrated. When converting to polar, you replace all instances of ``$x^2+y^2$" with ``$r^2$" and you replace ``$\mathrm{d}A$" with ``$r \mathrm{d}r\mathrm{d}\theta$" (not merely $\mathrm{d}r\mathrm{d}\theta$!! This is due to the so-called \href{https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant}{Jacobian matrix}.)
\begin{enumerate}[1.]
\item Let $R$ be the region defined by the circle of radius $2$. Compute $\displaystyle\iint_R x^2+y^2 \mathrm{d}A$. 
\item Let $R$ be the upper semicircle of radius $1$. Compute $\displaystyle\iint_R x^2+y^2 \mathrm{d}A$. 
\item Let $R$ be the quarter circle in the first quadrant of radius $3$. Compute $\displaystyle\iint_R \sqrt{x^2+y^2} \mathrm{d}A$.
\item Let $R$ be the annuluar region between the circles of radius $2$ and radius $5$. Compute $\displaystyle\iint_R x^2+y^2 \mathrm{d}A$. 
\item Let $R$ be defined by the circle of radius $1$. Compute $\displaystyle\iint_R (x^2+y^2)^2 + (x^2+y^2) \mathrm{d}A$.
\item Let $R$ be defined by the circle of radius $5$. Compute $\displaystyle\iint_R e^{x^2+y^2} \mathrm{d}A$. 

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