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\begin{center}
\textbf{Multiple Integration}

\textit{\textbf{Iteration of Double Integrals}}
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\textbf{Question}


Calculate the given iterated integral.

$$\int_0^2 \,dy \int_0^y y^2 e^{xy} \,dx$$


\textbf{Answer}

\begin{eqnarray*}
& & \int_0^2 \,dy \int_0^y y^2 e^{xy} \,dx\\
& = & \int_0^2y^2 \,dy \left ( \left. \frac{1}{y}e^{xy} \right
|_{x=0}^{x=y} \right )\\
& = & \int_0^2 y \left ( e^{y^2}-1 \right ) \,dy = \left
. \frac{e^{y^2}-y^2}{2} \right |_0^2\\
& = & \frac{e^4-5}{2}
\end{eqnarray*}


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