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\flushleft{\textbf{15.2}} \\

\vspace{10pt}

\begin{itemize}

\item Choosing the appropriate order of integration for an iterated integral \\
\vspace{10pt}
For almost all double integrals, we will need to compute them as iterated integrals. This requires choosing an order in which to integrate:
\[
\int \int_R f(x,y) \, dA = \left\{ \begin{array}{c}
\int_a^b \int_c^d f(x,y) \, dy \, dx \\ \\
\textrm{ or } \\ \\
\int_c^d \int_a^b f(x,y) \, dx \, dy \\
\end{array} \right.
\]
where $R = [a,b] \times [c,d]$. The following trick can help us figure out which order is easier. In the case of the top integral, $dy \, dx$, the first computation you will have to perform is to find
\[
\int f(x,y) \, dy
\]
where we will consider $x$ to be constant. In the case of the bottom integral, you will first have to compute
\[
\int f(x,y) \, dx
\]
where we will consider $y$ to be constant. To see which integral will be easier to compute at first, simply replace the constant variable with an actual constant. Usually a $1$. Just be warned that actually performing this integration can be misleading. And sometimes choosing a $1$ may trivialize the integral (for example, choosing $x = 1$ for an integrand involving $\ln x$ will give $ln 1 = 0$, not accurately representing the difficulty of the integral.)

\begin{enumerate}
\item 
\[
\int \int_R y \sin(xy) \, dA, \quad \quad R = \left[0,\pi/2 \right] \times \left[ 0, \pi/2\right]
\]
\begin{enumerate}
\item Write down the two iterated integrals.
\item Write down the inner integral of each, without bounds. \\
\vspace{10pt}
\textbf{Instructor:} Students should have
\[
\int y \sin(xy) \, dy \quad \textrm{ and } \quad \int y \sin(xy) \, dx \textrm{.}
\]
\item In the integral $dy$, replace $x$ with $1$. What method would you use to integrate this one variable integral?
\item In the integral $dx$, replace $y$ with $1$. What method would you use to integrate this one variable integral?
\item Is one of these methods easier? That is the first integral you want to do. That should be your inner integral.
\end{enumerate}

\item 
\[
\int \int_R \frac{\ln(xy)}{x^2y} \, dA, \quad \quad R = \left[1,2\right] \times \left[ 1, 2\right]
\]
\begin{enumerate}
\item Write down the two iterated integrals.
\item Write down the inner integral of each, without bounds. \\
\vspace{10pt}
\textbf{Instructor:} Students should have
\[
\int \frac{\ln(xy)}{x^2y} \, dy \quad \textrm{ and } \quad \int \frac{\ln(xy)}{x^2y} \, dx \textrm{.}
\]
\item In the integral $dy$, replace $x$ with $1$. What method would you use to integrate this one variable integral?
\item In the integral $dx$, replace $y$ with $1$. What method would you use to integrate this one variable integral?
\item Is one of these methods easier? That is the first integral you want to do. That should be your inner integral.
\end{enumerate}

\item 
\[
\int \int_R \frac{x}{\sqrt{1-x^2y^2}} \, dA, \quad \quad R = \left[1, \frac{1}{\sqrt{2}} \right] \times \left[ 1, \frac{1}{\sqrt{2}}\right]
\]
\begin{enumerate}
\item Write down the two iterated integrals.
\item Write down the inner integral of each, without bounds. \\
\vspace{10pt}
\textbf{Instructor:} Students should have
\[
\int \frac{x}{\sqrt{1-x^2y^2}} \, dy \quad \textrm{ and } \quad \int \frac{x}{\sqrt{1-x^2y^2}} \, dx \textrm{.}
\]
\item In the integral $dy$, replace $x$ with $1$. What method would you use to integrate this one variable integral?
\item In the integral $dx$, replace $y$ with $1$. What method would you use to integrate this one variable integral?
\item Is one of these methods easier? That is the first integral you want to do. That should be your inner integral.
\end{enumerate}

\end{enumerate}


\end{itemize}






\end{document}