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\begin{center}
\textbf{Multiple Integration}

\textit{\textbf{Iteration of Double Integrals}}
\end{center}

\textbf{Question}


Calculate the given double integral by iteration in the region
defined by the given curves.

$\ds \int\!\!\!\int_R xy^2 \,dA$

With $R$ being the region bounded by $y=x^2$ and $x=y^2$ in the first
quadrant.


\textbf{Answer}

\begin{eqnarray*}
\int\!\!\!\int_R xy^2 \,dA & = & \int_0^1 x \,dx \int_{x^2}^{\sqrt{x}}
y^2 \,dy\\
& = & \int_0^1 x \,dx \left. \left ( \frac{1}{3}y^3 \right ) \right
|_{y=x^2}^{y=\sqrt{x}}\\
& = & \frac{1}{3} \int_0^1 (x^{5/2} - x^7) \,dx\\
& = & \frac{1}{3} \left. \left (\frac{2}{7}x^{7/2} - \frac{x^8}{8}
\right ) \right |_0^1\\
=\frac{1}{3} \left ( \frac{2}{7} - \frac{1}{8} \right ) = \frac{3}{56}
\end{eqnarray*}

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