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Math Help - Disc method and shell method giving different solutions

  1. #1
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    Disc method and shell method giving different solutions

    I have the function f(x)=x2

    I am to give the volume of the shape when this function is spun 360 degrees around the x axis by using the disc method, and the shell method, from x=1 and x=3.

    When I work out the disc method, I get (242pi)/5. But when I use the shell method, I get (232pi)/5. I have been looking over my work, and it all seems to work out, so I am curious why they are different. I will show my setups below:

    Disc method:
    v=pi*r2*h
    r=y=f(x)=x2
    h=dx
    v=pi(x4) dx

    Then I integrate and get that v=pi[(1/5)x5]31 which gives me (242pi)/5.


    Shell method:
    v=2pi*r*h dy
    r=y
    h=3-f(y)=3-sqrt(y)
    v=2pi(3y-y3/2) dy

    I then integrate, and get that v=2pi[(3/2)y2-(2/5)y5/2]91 which then gives me (232pi)/5.

    Have I done something wrong here? Or is there a reason for the discrepancy?
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  2. #2
    MHF Contributor
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    Re: Disc method and shell method giving different solutions

    Check your limits for the shells
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  3. #3
    MHF Contributor
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    Re: Disc method and shell method giving different solutions

    Quote Originally Posted by Latsabb View Post
    Shell method:
    v=2pi*r*h dy
    r=y
    h=3-f(y)=3-sqrt(y)
    v=2pi(3y-y3/2) dy

    I then integrate, and get that v=2pi[(3/2)y2-(2/5)y5/2]91 which then gives me (232pi)/5.

    Have I done something wrong here? Or is there a reason for the discrepancy?
    As tom said, your limits of integration are wrong for the shell method. Additionally, the h you use is only correct on the limits you currently have for the integration.

    Hint: Currently, you are finding the volume of a shape that has a cylindrical hole through the center.
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  4. #4
    Junior Member
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    Bergen, Norway
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    Re: Disc method and shell method giving different solutions

    Yes, I worked it over, and saw my mistake, that I needed to take it from 0 to 9, and then subtract the small "spike" from the tip of the cone. Thank you for the hints.
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