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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 over the triangle defined by the vertices $(0,0)$, $(a,0)$
and $(0,b)$, and below the plane $z=2-(x/a)-(y/b)$.


\textbf{Answer}

\begin{eqnarray*}
V & = & \int\!\!\!\int_Y \left (2 - \frac{x}{a} - \frac{y}{b} \right )
\,dA\\
& = & \int_0^{b(1-(x/a))} \left ( 2 - \frac{x}{a} - \frac{y}{b} \right
) \,dy\\
& = & \int_0^a \left [ \left ( 2-\frac{x}{a} \right ) b \left ( 1
-\frac{x}{a} \right ) - \frac{1}{2b}b^2 \left ( 1-\frac{x}{a} \right
)^2 \right ] \,dx\\
& = & \frac{b}{a} \int_0^a \left ( 3 - \frac{4x}{a} + \frac{x^2}{a^2}
\right ) \,dx\\
& = & \frac{b}{2} \left. \left ( 3x - frac{2x^2}{a} + \frac{x}{3a^2}
\right ) \right |_0^a\\
& = & \frac{2}{3}ab \textrm{cu. units}
\end{eqnarray*}

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