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Find the volume V

  1. Jul 16, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the volume V of the solid inside both x^2 + y^2 + z^2 = 4 and x^2 + y^2 = 1.

    2. Relevant equations
    So I get how to set it up; you use cylindrical coordinates because it makes life a whole lot simpler BUT the answer is (4pi/3)(8-3^(3/2)) and I got (2pi/3)(8-3^(3/2)). So as you can see, I'm off by a factor of 2.

    3. The attempt at a solution
    The integrands I have are: z=(0,sqrt(4-r^2); theta = (0, 2pi) and r = (0,1). Since I'm off by a factor of 2, I'm thinking that for theta I should integrate from 0 to 4pi, but conceptually I don't get why.

    Any and all help would be much appreciated! :D
  2. jcsd
  3. Jul 16, 2011 #2


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    You just found the volume above the xy plane. There is an equal volume below the xy plane.
  4. Jul 16, 2011 #3
    I love this freaking forum!!! so would theta range from -2pi to 2pi ? I only did with respect to the positive axis. Or even simpler just multiply by 2 because I found the area for just half of the xy plane?
  5. Jul 16, 2011 #4


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    No about the theta; that isn't where the problem is. The problem is your z limit. You went from z = 0 to z on the top surface. You need to either change your lower limit to z on the bottom surface or double your answer because of the symmetry.
  6. Jul 16, 2011 #5
    THANK YOU SO MUCH! I now see hwere I went wrong. I just reattempted the problem using your advice and changed the z limits from (-sqrt(4-r^2),sqrt(4-r^2)) and got the answer I was supposed to get.

    Thanks again for your help and speedy response!
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