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Math Help - General question about cylindrical shell

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    General question about cylindrical shell

    I had this hw question (I solved to completion) Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by the following curves.
    Find V by cylindrical shell


    I understand that the formula is integral 2pi(radius)(height)d(x or y)

    To find the height, you do the "top of the shell"-"the bottom of the shell"
    in this problem, it was sqrtx-x^2

    My question is, why is the radius = to x?





    Is it that all cylindrical shell problems that the radius=x? My professor showed us numerous examples and they all had radius=x

    I made a note in my book that radius could be the line it touches in a graph
    (we had a problem that said to find volume revolving y axis by y=x^2 y=x ... Anyway, after making the cylindrical shell, the radius line we drew hit the line y=x so I figured maybe this was a rule??)

    I am confused how to find the radius. Any help appreciated! Thank you.

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    Re: General question about cylindrical shell

    The radius of an arbitrary shell is its distance from the axis of rotation. If you had rotated about the line x = -1, then the radius would be x - (-1) = x + 1.
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    Re: General question about cylindrical shell

    Quote Originally Posted by Steelers72 View Post
    I had this hw question (I solved to completion) Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by the following curves.
    Find V by cylindrical shell


    I understand that the formula is integral 2pi(radius)(height)d(x or y)

    To find the height, you do the "top of the shell"-"the bottom of the shell"
    in this problem, it was sqrtx-x^2

    My question is, why is the radius = to x?





    Is it that all cylindrical shell problems that the radius=x? My professor showed us numerous examples and they all had radius=x

    I made a note in my book that radius could be the line it touches in a graph
    (we had a problem that said to find volume revolving y axis by y=x^2 y=x ... Anyway, after making the cylindrical shell, the radius line we drew hit the line y=x so I figured maybe this was a rule??)

    I am confused how to find the radius. Any help appreciated! Thank you.

    You need to understand that the way volume by cylindrical shells works is that you fill up your region of rotation with cylinders, and so the volume ends up being numerically equal to the sum of all the surface areas of the cylinders.

    Since you are rotating about the y axis, the radius is VERTICAL, and so the radius will be the distance from the x-axis to the y-value, so y.

    The height of your cylinder is the difference in your x values. We can see that in this region, since \displaystyle  \sqrt{y} \geq y^2, the height will be \displaystyle \sqrt{y} - y^2.

    So the surface area of each cylinder will be \displaystyle 2\pi \, r\, h = 2\pi \, y \left( \sqrt{y} - y^2 \right) and so the volume is calculated by

    \displaystyle V = \int_0^1 { 2\pi\,y \left( \sqrt{y} - y^2 \right) dy }
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    Re: General question about cylindrical shell

    Thank you both. So, the height is difference of the two x values and the radius, in this case, was the distance from the x axis to the y value which= y.

    I really appreciate both of your help!
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