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Math Help - Finding the shell height in the Shell Method

  1. #1
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    Finding the shell height in the Shell Method

    Good afternoon.

    I am working on a problem very similar to the following.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Find the volume of the solid generated by revolving x=3-y^2, bounded by the x-axis, the y-axis, x=3 , and y=\sqrt{2}, about the x-axis.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Based on the information given in this problem, I have been able to set up the integral to show what is below.

    \int_0^3 2\pi r (3-y^2) dy

    My one problem is the r in the integral. I know how to obtain it when the region is being revolved around the y-axis, but I am not sure how to obtain it when the region is being revolved around the x-axis. Is it the distance from the x-axis to the top of the bounded region, or do you have to obtain it via another method?

    Thank you in advance for any assistance given.
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  2. #2
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    Quote Originally Posted by mathgeek777 View Post
    Good afternoon.

    I am working on a problem very similar to the following.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Find the volume of the solid generated by revolving x=3-y^2, bounded by the x-axis, the y-axis, x=3 , and y=\sqrt{2}, about the x-axis.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Based on the information given in this problem, I have been able to set up the integral to show what is below.

    \int_0^3 2\pi r (3-y^2) dy

    My one problem is the r in the integral. I know how to obtain it when the region is being revolved around the y-axis, but I am not sure how to obtain it when the region is being revolved around the x-axis. Is it the distance from the x-axis to the top of the bounded region, or do you have to obtain it via another method?

    Thank you in advance for any assistance given.
    r = distance from the x-axis to a representative shell ...
    r = y in this case.

    also, check your limits of integration ...

    V = 2\pi \int_0^{\sqrt{2}} y(3-y^2) \, dy

    note the attached graph.
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  3. #3
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    Actually, I misspoke when I stated the upper bound of y. It was actually y=\sqrt{3}, but it is irrelevant for the purposes of this question. Thanks for the help skeeter.
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