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\begin{center}
\textbf{Multiple Integration}

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


Calculate the given double integral by iteration.

$\ds \int\!\!\!\int_S (\sin x + \cos y) \,dA$

With $S$ being the square $o \le x,y \le \pi/2$.


\textbf{Answer}

\begin{eqnarray*}
& & \int\!\!\!\int_S (\sin x + \cos y) \,dA\\
& = & \int_0^{\pi/2} \,dx \int_0^{\pi/2} (\sin x + \cos y ) \,dy\\
& = & \int_0^{\pi/2} \,dx \left. (y\sin x + \sin y ) \right
|_{y=0}^{y=\pi/2}\\
& = & \int_0^{\pi/2} \left ( \frac{\pi}{2} \sin x +1 \right ) \,dx\\
& = & \left. \left ( - \frac{\pi}{2} \cos x + x \right ) \right
|_0^{\pi/2}\\
& = & \frac{\pi}{2} + \frac{\pi}{2} = \pi
\end{eqnarray*}

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