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\textbf{Multiple Integration}

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

Calculate the given double integrals by iteration

$\ds \int\!\!\!\int_T \sqrt{a^2 -y^2} \,dA$

With $T$ being the triangle with vertices $(0,0)$, $(a,0)$ and
$(a,a)$.


\textbf{Answer}

\begin{eqnarray*}
I & = & \int\!\!\!\int_T \sqrt{a^2 -y^2} \,dA\\
& = & \int_0^a \sqrt{a^2-y^2} \,dy \int_y^a \,dx\\
& = & \int_0^a (a-y) \sqrt{a^2-y^2} \,dy\\
& = & a\int_0^a \sqrt{a^2-y^2} \,dy - \int_0^a y\sqrt{a^2-y^2} \,dy\\
& & \textrm{Let }u=a^2-y^2\\
& & \textrm{and }du=-2ydy\\
& &\\ 
\Rightarrow I & = & a \frac{\pi a^2}{4} + \frac{1}{2}\int_{a^2}^0
u^{1/2} \,du\\
& = & \frac{\pi a^3}{4} - \left. \frac{1}{3} u^{3/2} \right |_0^{a^2}\\
& = & \left ( \frac{\pi}{4} - \frac{1}{3} \right ) a^4
\end{eqnarray*} 

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