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Math Help - Cylindrical shells revolved about the line y=1

  1. #1
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    Arrow Cylindrical shells revolved about the line y=1

    Use cylindrical shells to find the volume of the solid that is generate when the region that enclosed by y=x^3, y=1, x=0 is revolved about the line y=1

    As it said "cylindrical shells" to find the volume.

    I transform y=x^3 to x=\sqrt[3]{y}

    The area of this surface is 2\pi x f(x), and then integrate from a to b to get cylindrical shells formula. For this problem, I changed a to b to c to d. And 2\pi x f(x) to 2\pi y f(y)

    My problem is to be revolved about the line y=1, that means it similar to revolve about x-axis, but higher 1 unit. I don't know how to calculate it.


    I don't know how to find y and f(y).


    Please help, thank you.
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  2. #2
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    Am I correct, if the answer is 2\pi by using 2\pi (y+1)(1-\sqrt[3]{y}) as area.

    ????
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  3. #3
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    With the y+1, you are revolving about y=-1.

    With shells, the cross sections are parallel to the axis about which you are revolving. So, we integrate in terms of x.

    2{\pi}\int_{0}^{1}(1-x)x^{3}dx
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