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Negative Volume of Revolution?

  1. Jan 10, 2015 #1
    There is a nice equation made by Nobuo Yamamoto which describes the curve of an egg and it is:

    (x^2 + y^2)^2 = ax^3 + (3/10)xy^2, where a is the length of the major axis of the egg.

    Solve this equation for y, we get:

    y=+/- sqrt((3/20)ax - x^2 + xsqrt((7/10)ax + (9/400)a^2))

    When I rotate the function around the x-axis by 2(pi), the result is a negative volume of (-1)((pi)(a^3))/12.

    I don't know what I am doing wrong, or how I can fix this problem.

    Can someone please help me out on this?
     
  2. jcsd
  3. Jan 10, 2015 #2

    phinds

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    Show your work. How else might we see where you went wrong?
     
  4. Jan 10, 2015 #3
    Sorry, new to this site.

    I am using the positive case of the equation mentioned.
    1.png 2.png 3.png 4.png
    5.png
     
  5. Jan 12, 2015 #4

    vela

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    It looks like you didn't solve for ##y## correctly. I get
    $$y = \sqrt{\left(\frac b2\right)x-x^2+x\sqrt{\left(\frac b2\right)^2+(a-b)x}}$$ where ##b=3/10##.

    To integrate the term of the form ##x\sqrt{c+dx}##, try using the substitution ##u=c+dx##.
     
  6. Jan 12, 2015 #5
  7. Jan 13, 2015 #6

    vela

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    OK, so you have a typo in the equation in your first post. Your expression for ##y## is correct, but you didn't evaluate the third integral correctly.
     
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