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\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 & = & \int\!\!\!\int_R \frac{x}{y} e^y \,dA\\
& = & \int_0^1 \frac{e^y}{y} \,dy \int_y^{\sqrt{y}} x \,dx\\
& = & \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 & = & \frac{1}{2} \left [ \left. (1-y)e^y \right |_0^1 +
int_0^1e^y \,dy \right ]\\
& = & -\frac{1}{2} + \frac{1}{2}(e-1) = \frac{e}{2} -1
\end{eqnarray*}
\end{document}