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\textbf{Multiple Integration}

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


Calculate the given iterated integral.

$$\int_0^{\pi} \! \! \! \int_{-x}^x \cos y \,dy \,dx$$


\textbf{Answer}

\begin{eqnarray*}
& & \int_0^{\pi} \! \! \! \int_{-x}^x \cos y \,dy \,dx\\
& = & \int_0^{\pi} \left. \sin y \right |_{y=-x}^{y=x} \,dx\\
& = & 2 \int_0^{\pi} \sin x \,dx = -2 \left. \cos x \right |_0^{\pi}\\
& = & 4
\end{eqnarray*}

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