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Math Help - cylindrical shells about the x-axis almost got it

  1. #1
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    cylindrical shells about the x-axis almost got it

    cylindrical shells about the x-axis almost got it-question.jpg

    I'm having a hard time with this, I'm getting (804pi)/5 which I'm sure is wildly inaccurate

    which step is wrong?
    x = (y+2)^3 + 1
    new limits are 0 to 2
    integrate:
    2 pi (integral from 0 to 2){ ((y+2)^3 +1)(y) dy }
    where y is the radius and (y+2)^3 +1 is the height
    answer = (804pi)/5

    there's a very similar question in the textbook with nearly the same numbers that comes out to like (81pi)/5 so I know I'm on the right track
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: cylindrical shells about the x-axis almost got it

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

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ID:	22391

    I'm having a hard time with this, I'm getting (804pi)/5 which I'm sure is wildly inaccurate

    which step is wrong?
    x = (y+2)^3 + 1
    new limits are 0 to 2
    integrate:
    2 pi (integral from 0 to 2){ ((y+2)^3 +1)(y) dy }
    where y is the radius and (y+2)^3 +1 is the height
    answer = (804pi)/5

    there's a very similar question in the textbook with nearly the same numbers that comes out to like (81pi)/5 so I know I'm on the right track
    Why do you think your answer is wrong?
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    Re: cylindrical shells about the x-axis almost got it

    Quote Originally Posted by alexmahone View Post
    Why do you think your answer is wrong?
    well I tried the example in the book (I know that answer is right because it is given) but couldn't get that answer either
    also this homework problem gets handed in online and the site instantly tells you whether or not it's right
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: cylindrical shells about the x-axis almost got it

    Quote Originally Posted by RamzaBeoulve View Post
    well I tried the example in the book (I know that answer is right because it is given) but couldn't get that answer either
    also this homework problem gets handed in online and the site instantly tells you whether or not it's right
    Your solution seems right to me.
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    Re: cylindrical shells about the x-axis almost got it

    Quote Originally Posted by alexmahone View Post
    Your solution seems right to me.
    would you mind doing another similar problem and reporting the answer you get?
    if we both get the same answer, then I suppose the website is wrong and we're right.

    obtain the volume using cylindrical shells of the following function when rotated about the x-axis
    y=(x^(1/3)) -2
    8<=x<=27
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    Re: cylindrical shells about the x-axis almost got it

    V = 2\pi \int_0^2 y[64-(y+2)^3] \, dy = \frac{496\pi}{5}

    confirmed the solution using disks ...

    V = \pi \int_9^{65} (\sqrt[3]{x-1}-2)^2 \, dx = \frac{496\pi}{5}
    Attached Thumbnails Attached Thumbnails cylindrical shells about the x-axis almost got it-22391d1316899128-cylindrical-shells-about-x-axis-almost-got-question.jpg  
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    MHF Contributor alexmahone's Avatar
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    Re: cylindrical shells about the x-axis almost got it

    Quote Originally Posted by skeeter View Post
    V = 2\pi \int_0^2 y[64-(y+2)^3] \, dy
    How did you get the term inside the square brackets? I can see that it is equal to 65-x but why is it not simply x?
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    Re: cylindrical shells about the x-axis almost got it

    we have almost the same formula except you have a (64-height) in yours
    where does that come from?
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    Re: cylindrical shells about the x-axis almost got it

    Quote Originally Posted by alexmahone View Post
    How did you get the term inside the square brackets? I can see that it is equal to 65-x but why is it not simply x?
    because integration is w/r to y

    y = \sqrt[3]{x-1} - 2

    y+2 = \sqrt[3]{x-1}

    (y+2)^3 = x-1

    x = (y+2)^3 + 1

    horizontal "height" of the shell = right x-value - left x-value ...

    65 - x = 65 - [(y+2)^3 + 1] = 64 - (y+2)^3
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  10. #10
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    Re: cylindrical shells about the x-axis almost got it

    obtain the volume using cylindrical shells of the following function when rotated about the x-axis
    y=(x^(1/3)) -2
    8<=x<=27
    shells ...

    V = 2\pi \int_0^1 y[27 - (y+2)^3] \, dy

    confirm the integral by using disks about the x-axis ...

    V = \pi \int_8^{27} (\sqrt[3]{x} - 2)^2 \, dx
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