(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate the integral

integral from 0 to 1 of (integral from y to sqrt(2-y^2) of (3(x-y))dxdy)

by converting to polar coordinates.

2. Relevant equations

x = rcos(theta)

y = rsin(theta)

3. The attempt at a solution

By drawing a picture of the bounds, I concluded that r goes from 0 to sqrt(2) and theta goes from pi/4 to pi/2.

I tried to integrate like this:

integral from 0 to sqrt(2) of 3(rcos(theta) - rsin(theta) r dr dtheta

= integral from 0 to sqrt(2) of 3r^2 (cos(theta) - sin(theta)) dr dtheta

= r^3(cos(theta) - sin(theta)) where r is from 0 to sqrt(2)

=2sqrt(2) * integral from pi/4 to pi/2 of (cos(theta)-sin(theta)) dtheta

=2sqrt(2) * (sin(theta)+cos(theta)) with theta from pi/4 to pi/2

=2sqrt(2) *((1+0)-((sqrt(2)/2) + (sqrt(2)/2))

=2sqrt(2) *(1-sqrt(2))

=2sqrt(2) - 4

The actual answer is 4 - 2sqrt(2) so I missed a sign somewhere, but I cannot find it. Sorry if it's difficult to read. Any help would be greatly appreciated.

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# Homework Help: Evaluate the integral by converting to polar coordinates

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