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Math Help - Integration with Cylindrical shells

  1. #1
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    Wink Integration with Cylindrical shells

    Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

    The question:
    Find the volume of S using the method of cylindrical shells.

    S is generated by rotating about the x-axis the region bounded by
    y = x^2 , y = 0, and x = 1

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  2. #2
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    Quote Originally Posted by Fredrik373 View Post
    Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

    The question:
    Find the volume of S using the method of cylindrical shells.

    S is generated by rotating about the x-axis the region bounded by
    y = x^2 , y = 0, and x = 1

    Since are rotating around the x-axis, and want to use "cylindical shells", you will want to integrate with resect to y.

    The integral will be \int_0^1 2\pi r h dy where "r" is the radius of the cylinder- that will be just "y", the distance from the x-axis to the line that, rotated around the x-axis, creates the cylinder, and "h" is the length of cylinder, the length of the line that creates the cylinder- that will be "1- x" where x is the x-coordinate of the point (x, y) at the left end of that line: since y= x^2, x= ?
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  3. #3
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    Quote Originally Posted by Fredrik373 View Post
    Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

    The question:
    Find the volume of S using the method of cylindrical shells.

    S is generated by rotating about the x-axis the region bounded by
    y = x^2 , y = 0, and x = 1
    shells w/r to y ...

    V = 2\pi \int_0^1 y(1 -  \sqrt{y}) \, dy


    disks w/r to x ...

    V = \pi  \int_0^1 x^4 \, dx
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