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Homework Help: Another integration in polar

  1. Jul 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Compute the indicated solid in POLAR COORDINATE using double integrals.

    Below z = 4 - x^2 - y^2, z = x^2 + y^2, between y = x and y = 0.


    2. Relevant equations
    3. The attempt at a solution

    First of all, the integrand is z = 4 - x^2-y^2 which in polar is 4 - r^2

    The limit for the region D in polar is the intersections of y = x, y = 0 of the circle. To find that particular circle I think we have to solve the two z equations, which give us x^2 + y^2 = 2 in the end. This is a circle with radius 2

    The limit of region D is 0 <= r <= sqrt(2), and for theta (i use x) is 0 <= x < pi/6
    I am not sure whether pi/6 is really the intersecting point of y = x on the circle.... Please cofirm that...

    This will give us the double integrals
    integral (0 to pi/6) integral (0 to sqrt(2) (4 - r^2)r dr d theta

    I think this give us pi/2 which is right from the book. But the book only gave pi/2 there is no work shown so I can't tell whether my work is right or not.

    Please tell me if I am wrong in the limit of integrations.
    Thankyou.
     
  2. jcsd
  3. Jul 10, 2010 #2
    Are you sure it's not below z = 4 - r^2...and above z = r^2? Because It doesn't make much sense if it's below both.

    mhmm...good so far.

    No, sorry this doesn't work. First of all, x^2 + y^2 = 2 is a circle with radius sqrt(2), not 2. Regardless of that, however, you are not finding your region D correctly. The region D is the bounded by the trace of the paraboloid z=4-r^2 onto the xy-plane; i.e., where z=0. Can you figure out what type of curve the paraboloid traces here?

    The region D is pretty. The limits of r are from 0 to some integer (which I'll leave you to figure out from my notes above). The problem asks for the volume bounded by the two paraboloids between the planes y=x and y=0. You use these two equations to find the limits for theta. Once again, you look at the xy-plane where your region D lies. What is theta when y=0? What is theta when y=x? These two values become your limits for theta.

    And, yes, the final answer is pi/2.

    I hope this helps. Good luck.
     
  4. Jul 10, 2010 #3

    benorin

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    Homework Helper

    Your integrand lacks the lower boundary, and the angle between y=0 and y=x is Pi/4. Fixing these gives Pi/2 as well, the correct answer.
     
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