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\textbf{Multiple Integration}

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

Find the volume for the solid defined by 

The space inside two cylinders, $x^2+y^2=a^2$ and $y^2+z^2=a^2$.


\textbf{Answer}

\begin{eqnarray*}
V & = & 8 \times (\textrm{vol in first octant})\\
& = & 8 \int_0^a \,dx \int_0^{\sqrt{a^2-x^2}} \sqrt{a^2-x^2} \,dy\\
& = & 8\int_0^a (a^2-x^2)\,dx\\
& = & 8 \left. \left ( a^2x - \frac{x^3}{3} \right ) \right |_0^a\\
& = & \frac{16}{3}a^3 \textrm{cu. units}
\end{eqnarray*}

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