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\begin{center}
\textbf{Multiple Integration}

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

Find the volume of the given solid

Below $z=1-x^2-y^2$ and over the region $x \ge 0$, $y \ge 0$, $x+y \le
1$.


\textbf{Answer}

\begin{eqnarray*}
V & = & \int_0^1 \,dx \int_0^{1-x} (1-x^2-y^2) \,dy\\
& = & \int_0^1 \left. \left ( (1-x^2)y-\frac{y^3}{3} \right ) \right
|_{y=0}^{y=1-x}\,dx\\
& = & \int_0^1 \left ( (1-x^2)(1-x) - \frac{(1-x)^3}{3} \right )
\,dx\\
& = & \int_0^1 \left ( \frac{2}{3} - 2x^2 + \frac{4x^3}{3}\right ) \,dx\\
& = & \frac{2}{3} - \frac{2}{3} + \frac{1}{3} = \frac{1}{3}
\textrm{cu. units}
\end{eqnarray*}

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