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\textbf{Multiple Integration}

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


Calculate the given iterated integral.

$$\int_0^1 \! \! \! \int_0^y (xy+y^2) \,dx \,dy$$ 


\textbf{Answer}

\begin{eqnarray*}
& & \int_0^1 \! \! \! \int_0^y (xy+y^2) \,dx \,dy\\
& = & \int+0^1 \left. \left ( \frac{x^2y}{2} + xy^2 \right ) \right
|_{y=0}^{y=\pi/2} \,dy\\
& = & \frac{3}{2} \int_0^1 y^3 \,dy = \frac{3}{8}
\end{eqnarray*}

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