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Math Help - Integration Query

  1. #1
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    Integration Query

    Hi,
    I was looking through some past papers and came across the following:

    Express the volume that is contained inside both the sphere x^2+y^2+z^2=4 and the cylinder x^2+y^2=1 as a triple integral in:

    1. Cartesian Co-ordinates.
    2. Cylindrical Polar Co-ordinates.

    Then, I am asked to evaluate (2) to find the required volume.

    I am unsure how to go about answering this. I thought about writing:

    z=sqrt(4-x^2-y^2).

    Then, I would have a triple integral. I would integrate with respect to z first, between -sqrt(4-x^2-y^2) and +sqrt(4-x^2-y^2). Then, y between -sqrt(1-x^2) and +sqrt(1-x^2), finally x between -2 and 2. However, should I be integrating x^2+y^2-1 here?!?

    I can see how the second part helps (r between 0 and 1, theta between 0 and two Pi), but I dont know how to go about writing it without the first part. I also dont know what z would be in this case.

    Any help would be appreciated, i've been scratching my head on this one the last week or so now!
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  2. #2
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    I take it you mean,"the solid that is bounded above and below by the sphere x^{2}+y^{2}+z^{2}=4 and inside the cylinder x^{2}+y^{2}=1".

    Polar:

    2\int_{0}^{2\pi}\int_{0}^{1}\int_{0}^{\sqrt{4-r^{2}}}rdzdrd{\theta}
    Last edited by galactus; April 30th 2008 at 07:17 AM.
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  3. #3
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    Quote Originally Posted by galactus View Post
    I take it you mean,"the solid that is bounded above and below by the sphere x^{2}+y^{2}+z^{2}=4 and inside the cylinder x^{2}+y^{2}=1".

    Polar:

    2\int_{0}^{2\pi}\int_{0}^{1}\int_{0}^{\sqrt{2-r^{2}}}rdzdrd{\theta}
    Hi,
    Thanks for your reply. I put the question down exactly as I was asked, the first part of the question asked me to express (x,y,z) in terms of (r, theta, z), and then this was the second part of the question. I dont understand how you got to the spherical part from the polar part. I can see that r will be from 0 to 1, and theta will be from 0 to 2Pi (just by the fact we are looking at a circle and a sphere) however I dont understand how you obtained the z limits. I can see that by re-writing the sphere x^2+y^2+z^2=4 as:

    z=sqrt(4-r^2)

    So how come you have the limit of integration as sqrt(2-r^2)? I can see where the r comes from, the Jacobian. However, where does the '2' come from at the start of the integral?

    Also, did I construct the problem in Cartesian Co-ordinates correctly?!?
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  4. #4
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    That 2 is a typo. Sorry, should be 4 as you pointed out.
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