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{\bf Exam Question

Topic: Double Integral}

The region $R$ in the $x$-$y$ plane is specified by the conditions
$$1\le x^2+y^2\le 9, \ x\ge 0.$$

Evaluate the double integral $$ \int \!\!\! \int_R
\sin\left(\frac{\pi(x^2+y^2)}{4}\right)\, d(x,y).$$

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{\bf Solution}

This integral is clearly to be done in polar coordinates
$$\int_{\theta=-\pi/2}^{\pi/2}\,
d\theta\int_{r=1}^3\sin\left(\frac{\pi(x^2+y^2)}{4}\right).r\, dr
$$ $$=\pi\left[-\frac{2}{\pi}\cos\left(\frac{\pi
r^2}{4}\right)\right]_1^3=2\left[-\cos\left(\frac{9\pi}{4}\right)
+\cos\left(\frac{\pi}{4}\right)\right]=0.$$



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