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\textbf{Multiple Integration}

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


Calculate the given iterated integral.

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


\textbf{Answer}

\begin{eqnarray*}
& & \int_0^1 \,dx \int_0^x (xy+y^2) \,dy\\
& = & \int_0^1\,dx \left. \left ( \frac{xy^2}{2} + \frac{y^3}{3}
\right ) \right |_{y=0}^{y=x}\\
& = & \frac{5}{6} \int_0^1 x^3 \,dx = \frac{5}{2}
\end{eqnarray*}


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