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Math Help - Find the volume inside a cone and a cylinder using change of variables

  1. #1
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    Find the volume inside a cone and a cylinder using change of variables

    I have been trying to figure this question out for a while with no luck.
    I have to find volume inside the cone z=2a−sqrt(x^2+y^2)and inside the cylinder x^2+y^2=2ay.
    I am trying to use cylindrical coordinates but I am just having some issues.

    If anyone could help I would appreciate it.

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  2. #2
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    Re: Find the volume inside a cone and a cylinder using change of variables

    What are the issues? Are they with

    \int_0^{2\pi}\ \int_0^{a \sin \theta}\ \int_0^{2a - r}\ r\ dz\ dr\ d\theta

    or thereafter?
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  3. #3
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    Re: Find the volume inside a cone and a cylinder using change of variables

    Quote Originally Posted by tom@ballooncalculus View Post
    What are the issues? Are they with

    \int_0^{2\pi}\ \int_0^{a \sin \theta}\ \int_0^{2a - r}\ r\ dz\ dr\ d\theta

    or thereafter?
    Yes my issues are with the above statement. I understand how you got boundaries for the integral however when I did it I got an answer of a^3*pi which is not the answer in the back of the book. The answer in the back of the book is a^3(2pi-32/9)... Now I don't know if I computed the integral incorrectly or what but if you could confirm the calculation it would be a huge help.
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  4. #4
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    Re: Find the volume inside a cone and a cylinder using change of variables

    Typo, we meant...

    \int_0^{2\pi}\ \int_0^{2a \sin \theta}\ \int_0^{2a - r}\ r\ dz\ dr\ d\theta

    ... yes? But I agree with your calculation (now multiplyied by 4). So there may be a difference of geometric interpretation? Hmm...
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