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How do you find the centroid of this?

  1. Apr 26, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the centroid of the shape formed by the equation y2=x3-x4, the x-axis, and the y-axis.

    2. Relevant equations
    A=∫f(x)dx
    Mx=∫(1/2)[f(x)]2dx
    My=∫x[f(x)]dx

    3. The attempt at a solution
    I'm stuck on the integral.
    I attempted u-substitution and got du=(1/2)(x3-x4)-1/2dx; "parts," trigonometric substitution/identities, and partial fractions don't seem to apply.
     
  2. jcsd
  3. Apr 26, 2016 #2

    SammyS

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    Try polar coordinates.
     
  4. Apr 26, 2016 #3
    I haven't worked in those for a while (I'm not sure I remember how to use them); but I'm fairly certain my teacher wants me to stick with x-y coordinates.
     
  5. Apr 26, 2016 #4

    SteamKing

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    You've already tried trig substitutions. Think of polar coordinates as a form of trig substitution.

    Besides, what will your teacher prefer? That you kept working with cartesian coordinates and didn't solve the problem, or you converted to polar coordinates and got an answer? :wink:
     
  6. Apr 27, 2016 #5

    vela

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    You swapped ##M_x## and ##M_y##.

    Try starting like this:
    $$\int x\sqrt{x^3-x^4}\,dx = \int x^2\sqrt{x-x^2}\,dx = \int x^2\sqrt{\frac 14-\left(x-\frac 12\right)^2}\,dx.$$ The last step comes from completing the square. Then try a few more substitutions and see if you get anywhere.
     
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