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Math Help - Cyndrical shells for volume of a sphere

  1. #1
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    Cyndrical shells for volume of a sphere

    I have to use the cylindrical shells formula to find the volume of a sphere of radius r. I wasn't sure on how to solve it, so I checked my answer guide for help. I'm confused on: how did the positive 2s from the first step turn into negative 2s in the second step? And on the third step, the (-2x) disappeared? What was solved to make these steps possible?

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    Quote Originally Posted by !!! View Post
    I have to use the cylindrical shells formula to find the volume of a sphere of radius r. I wasn't sure on how to solve it, so I checked my answer guide for help. I'm confused on: how did the positive 2s from the first step turn into negative 2s in the second step? And on the third step, the (-2x) disappeared? What was solved to make these steps possible?

    That is because the derivative of the (r^2 -x^2) is -2x dx
    You have only x dx in the original integral, so, in order for you to differentiate the integrand, the x dx must become -2x dx.
    So you multiply and divide at the same time the x dx by -2.

    In the integration,
    V = (-4pi/3)[(r^2 -x^2)^(3/2)]|(0 to r),
    V = (-4pi/3)[(r^2 -r^2)^(3/2) -(r^2 -0)^(3/2)]
    V = (-4pi/3)[(0) -(r^2)^(3/2)]
    V = (-4pi/3)[-r^3]
    And so the negatives disappeared again.
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