1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
  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


    User Avatar
    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Another integration polar Date
Another Improper Integral Using Complex Analysis Apr 25, 2017
Another Green's function Dec 21, 2016
Another simple double integral Jun 18, 2016
Another triple integral problem Feb 15, 2016
Another Polar Double Integral Nov 24, 2006