Evaluate the given integral by changing to polar coordinates.

Thanks in advance.

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- March 21st 2009, 08:47 AMBiscaima double integral
Evaluate the given integral by changing to polar coordinates.

Thanks in advance. - March 21st 2009, 09:46 AMOpalg
The integral is . You need to work out the region over which the integral takes place, and for that you must

**draw a picture**.

Notice that y goes from 0 to . The upper limit of that interval occurs when . Square both sides of that equation, rearrange it a bit, and you see that it can be written as . You will recognise that as the equation of the circle centred at (1,0) with radius 1. But the lower limit for y is 0, so the region of integration is the upper half of that circle.

That is stage 1 of the solution. Stage 2 is to describe that region in terms of polar coordinates.**Have you drawn a picture of that semicircular region yet?**If so, you should be able to see that goes from 0 to and, for each fixed value of , r goes from 0 to .

Once you have got that far, you can write the integral as , and the rest should be easy.