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Math Help - Volume of a Solid Enclosed by a Region

  1. #1
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    Volume of a Solid Enclosed by a Region

    Find the volume formed by rotating the region enclosed by the following functions about the y-axis.
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  2. #2
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    The curves intersects at (0,0) and (\sqrt{3},3\sqrt{3}), so using cilindrical shel method, the volume is given by V=2\pi\int_0^{3\sqrt{3}}x\left(\sqrt[3]{x}-\frac{x}{3}\right)dx.

    You may also see the problem like rotating the area between y=3x and y=x^3 about the x-axis from (0,0) to (\sqrt{3},3\sqrt{3}).
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  3. #3
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    intersection values ...

    0 = y^3 - 3y

    0 = y(y + \sqrt{3})(y - \sqrt{3})<br />

    two symmetrical regions in quads I and III

    using the method of washers and taking advantage of symmetry ...

    V = \pi \int_0^{\sqrt{3}} (3y)^2 - (y^3)^2 \, dy
    Last edited by skeeter; February 6th 2009 at 01:36 PM. Reason: fixed integral ... y is greater than or equal to 0 ... mea culpa.
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  4. #4
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    Skeeter, your "dy" version of the volume problem is wrong. The integral isn't supposed to have "2" as a constant at the left.....
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  5. #5
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    Quote Originally Posted by Kaitosan View Post
    Skeeter, your "dy" version of the volume problem is wrong. The integral isn't supposed to have "2" as a constant at the left.....
    I beg to differ ... graph x = y^3 and x = 3y and see for yourself.
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  6. #6
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    I'm having trouble integrating. Should the answer be 27.98433302?
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  7. #7
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    Quote Originally Posted by skeeter View Post
    I beg to differ ... graph x = y^3 and x = 3y and see for yourself.
    I'm telling you, don't embarrass yourself. You know how to use the calculator for definite integrals right? Try using Abu-Khali's "dx" version and compare it to your own "dy" version. Then try using my own "dy" version and you will see that you are wrong, I promise you.

    Edit: Sorry if I sounded like I was flaming you, that's certainly not my intention. It's just that I know I'm right, factually. Try have an open mind.

    2nd Edit: yes, pony, you're right, that's the right answer.
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  8. #8
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    my mistake ... I did not pay attention to y \geq 0
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