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Rina8888 [55]
3 years ago
12

Convert the integral below to polar coordinates and evaluate the integral.

Mathematics
1 answer:
White raven [17]3 years ago
7 0

I suppose the integral is

\displaystyle\int_0^5\int_y^{\sqrt{25-y^2}} xy\,\mathrm dx\,\mathrm dy

The integration region corresponds to a sector of a cirlce with radius 5 subtended by a central angle of π/4 rad. We can capture this region in polar coordinates by the set

R=\left\{(r,\theta)\mid0\le r\le5,0\le\theta\le\dfrac\pi4\right\}

Then x=r\cos\theta, y=r\sin\theta, and \mathrm dx\,\mathrm dy=r\,\mathrm dr\,\mathrm d\theta. So the integral becomes

\displaystyle\iint_Rxy\,\mathrm dx\,\mathrm dy=\int_0^{\pi/4}\int_0^5r^3\sin\theta\cos\theta\,\mathrm dr\,\mathrm d\theta=\frac{625}{16}

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