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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

$\ds \int\!\!\!\int_T \frac{xy}{1+x^4} \,dA$

With $T$ being the triangle with vertices $(0,0)$, $(1,0)$ and
$(1,1)$.


\textbf{Answer}

\begin{eqnarray*}
I & = & \int\!\!\!\int_T \frac{xy}{1+x^4} \,dA\\
 & = & \int_0^1 \frac{x}{1+x^4} \,dx \int_0^x y \,dy\\
& = & \frac{1}{2} \int_0^1 \frac{x^3}{1+x^4} \,dx\\
& = & \frac{1}{8} \left. \ln (1+x^4) \right |_0^1 = \frac{\ln 2}{8}
\end{eqnarray*}

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