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{\bf Exam Question

Topic: Double Integral}

$T$ is the triangle in the $x$-$y$ plane with vertices $(0,0),\
(0,3),\ (1,3).$

Evaluate the integral $$\int \!\!\! \int_T 6\exp(-y^2)\, d(x,y).$$

Express your answer in terms of e, and also as an approximation
correct to 6 decimal places using your calculator. \vspace{0.5in}

{\bf Solution}

The correct choice of order of integration has to be made. It
can't be done the other way round

\begin{eqnarray*}
I&=& \int_0^3\, dy\int_0^{y/3} 6\exp(-y^2)\, dx = \int_0^3
2y\exp(-y^2)\, dy \\ &=& \left[-exp\left(-y^2\right)\right]_0^3
=1-\mathrm{e}^{-9}=0.99877 \mathrm{\ \ (6\ d.p.)}
\end{eqnarray*}



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