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Evaluate the integral by converting to polar coordinates

  1. May 2, 2010 #1
    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.
     
  2. jcsd
  3. May 2, 2010 #2

    Mark44

    Staff: Mentor

    Here's your problem. The region of integration is the sector of a circle that extends from theta = 0 to theta = pi/4. The radius is as you have it.
     
  4. May 2, 2010 #3
    That solves the problem! Thank you very much!
     
    Last edited: May 2, 2010
  5. May 2, 2010 #4

    Mark44

    Staff: Mentor

    Sure, you're welcome. As you found out, it's very important to have an accurate understanding of the region of integration. In many problems of this type, the hardest part can often be representing the region of integration in a different coordinate system.
     
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