<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\newcommand{\un}{\underline}
\newcommand{\ds}{\displaystyle}
\begin{document}
\parindent=0pt

\begin{center}
\textbf{Multiple Integration}

\textit{\textbf{Iteration of Double Integrals}}
\end{center}

\textbf{Question}

Calculate the given double integrals by iteration

$\ds \int\!\!\!\int_R \frac{x}{y} e^y \,dA$

With $R$ being the region $0 /le x /le 1$, $x^2 /le y /e x$.


\textbf{Answer}

\begin{eqnarray*}
I &amp; = &amp; \int\!\!\!\int_R \frac{x}{y} e^y \,dA\\
&amp; = &amp; \int_0^1 \frac{e^y}{y} \,dy \int_y^{\sqrt{y}} x \,dx\\
&amp; = &amp; \frac{1}{2} \int_0^1 (1-y_e^y \,dy
\end{eqnarray*}
$$U=1-y \ \ \ \ \ dV=e^ydy$$
$$dU=-dy \ \ \ \ \ V=e^y$$
\begin{eqnarray*}
\Rightarrow I &amp; = &amp; \frac{1}{2} \left [ \left. (1-y)e^y \right |_0^1 +
int_0^1e^y \,dy \right ]\\
&amp; = &amp; -\frac{1}{2} + \frac{1}{2}(e-1) = \frac{e}{2} -1
\end{eqnarray*}
\end{document}



</pre></body></html>