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Thread: Volume of Revolution (Integration)

  1. #1
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    Volume of Revolution (Integration)

    Volume of Revolution (Integration)-capture.png

    Having a bit of trouble with this problem. Not with the setting up, but trying to solve the integral.

    I tried using the disk method, and cant integrate pi * integral(1/(x^2+1)^2 dx.

    also tried using the shell method, but cant seem to integrate 2pi* integral(y(sqrt(1/y-1))
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    Re: Volume of Revolution (Integration)

    Quote Originally Posted by lc99 View Post
    Click image for larger version. 

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    Having a bit of trouble with this problem. Not with the setting up, but trying to solve the integral.
    I tried using the disk method, and cant integrate pi * integral(1/(x^2+1)^2 dx.
    Look at this.
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    Re: Volume of Revolution (Integration)

    What method would i use to find the integral?
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    Re: Volume of Revolution (Integration)

    Quote Originally Posted by lc99 View Post
    What method would i use to find the integral?
    Open this PDF file.
    Integrate[ pi (1+x^(2))^(-2)dx,0,1] - Wolfram_Alpha.pdf
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    Re: Volume of Revolution (Integration)

    Quote Originally Posted by lc99 View Post
    What method would i use to find the integral?
    Try $x=\tan \theta,~dx = \sec^2\theta d\theta,~1+x^2=\sec^2\theta$.
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    Re: Volume of Revolution (Integration)

    Use the disk method



    take a small segment dx, and when you revolve the indicated region around the x-axis, the small segment will become a disk.

    Therefore, you need the volume of a disk which is pi x r^2 x h, where r is the radius of the disk and h is the height.

    if you look at the region, you will see that the radius = f(x) = 1/(x^2 + 1) while the height is dx.

    And the integral will become

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  7. #7
    Junior Member Walagaster's Avatar
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    Re: Volume of Revolution (Integration)

    @joshuaa: Didn't the OP indicate he already knows that and just wants to know the technique to work it out?
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    Re: Volume of Revolution (Integration)

    Quote Originally Posted by Walagaster View Post
    @joshuaa: Didn't the OP indicate he already knows that and just wants to know the technique to work it out?
    yeah, I just realized that. maybe, I should read the thread carefully next Time. thanks for the reminder!

    this integral is very interesting. I might post the whole steps of solving it later. And that will be my method!
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    Re: Volume of Revolution (Integration)



    use u substitution



















    be careful here, you cannot substitute the integral limits 0 to 1 here because u is not the original variable. so, get x back then substitute!

















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    Re: Volume of Revolution (Integration)

    Thank you. This is beautiful. You made it seem simple! Didn't think of trig sub here.
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    Re: Volume of Revolution (Integration)

    Quote Originally Posted by lc99 View Post
    Thank you. This is beautiful. You made it seem simple! Didn't think of trig sub here.
    The PDF that Plato offered is IMHO more readable.
    Thanks from Plato and topsquark
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