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\textbf{Multiple Integration}

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

Sketch the domain of integration, and calculate the iterated integral
for
$$\int_0^1 \,dx \int_x^1 \frac{y^{\lambda}}{x^2+y^2} \,dy \ \ \ \
(\lambda>0)$$


\textbf{Answer}

$$ \ $$
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\begin{eqnarray*}
I & = & \int_0^1 \,dx \int_x^1 \frac{y^{\lambda}}{x^2+y^2} \,dy \ \ \ \
(\lambda>0)\\
& = & \int\!\!\!\int_R \frac{y^{\lambda}}{x^2+y^2} \,dA\\
& = & \int_0^1 y^{\lambda} \,dy \int_0^y \frac{dx}{x^2+y^2}\\
& = & \int_0^1 \left. y^{\lambda} \,dy \frac{1}{y} \left ( \tan^-1
\frac{x}{y} \right ) \right |_{x=0}^{x=y}\\
& = & \frac{\pi}{4} \int_0^1 y^{\lambda -1} \,dy\\
& = & \left. \frac{\pi y^{\lambda}}{4\lambda} \right |_0^1 =
\frac{\pi}{4\lambda}
\end{eqnarray*}

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