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\textbf{Multiple Integration}

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


Calculate the given double integral by iteration.

$\ds \int\!\!\!\int_R (x^2 + y^2)\,dA$

With $R$ being the area $0 \le x \le a$, $0 \le y \le b$.


\textbf{Answer}

\begin{eqnarray*}
\int\!\!\!\int_R (x^2 + y^2)\,dA & = & \int_0^a \,dx \int_0^b
(x^2+y^2) \,dy\\
& = & \int_0^a \,dx \left. \left ( x^2y + \frac{y^3}{3} \right )
\right |_{y=0}^{y=b}\\
& = & \int_0^a \left ( bx^2 + \frac{1}{3}b^3 \right ) \,dx\\
& = & \frac{1}{3} \left | (bx^3 + b^3x) \right |_0^a\\
& = & \frac{1}{3}(a^3b+ab^3)
\end{eqnarray*}

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