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Homework Help: Difficult Centroid Problem

  1. May 31, 2004 #1
    This is a problem from a past final calculus test in my university (I'm studying for the finals which will come in around 1 week :) ):

    Find the centroid of a plane area in the first quadrant bounded by



    and the x axis.

    I tried splitting the plane into 2 parts, one from x = 0 to x = 2 (call this plane 1), and the other from x = 2 to x = 3 (call this plane 2). My tactic is to find the centroids of each plane. I can then easily find the centroid of the composite plane. Then I define these equations:

    [itex]y_1 = \sqrt{4-\frac{4}{9}x^2}[/itex]
    [itex]y_2 = \left({2^{2/3}-x^{2/3}}\right)^{3/2}[/itex]

    Here are the integrals I formulated:

    To find the area of plane 1:


    To find the area of plane 2:


    To find the (first) moment of plane 1 with respect to the y axis:


    To find the moment of plane 1 with respect to the x axis:


    And similiarly,


    I'm stuck at finding the area. The definite integral which I must evaluate is:


    The first term evaluates to (If I had done it correctly)


    On the other hand, I have no idea how to integrate the second term. Can anyone give me a hint?

    Or maybe, the method that I have chosen (dividing it into 2 plane, etc etc) results in a complex calculation. Is there any easier alternatives?

    Thanks a lot!!!
  2. jcsd
  3. May 31, 2004 #2


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    I don't think you did the first integration correctly.

    [tex]\int _0 ^2 \sqrt{4 - \frac{4}{9}x^2} dx[/tex]
    [tex] = 2\int _0 ^2 \sqrt{1 - {\left ( \frac{x}{3} \right ) }^2} dx[/tex]
    [tex] = 6\int_{x=0} ^{x=2} \cos ^2 \theta d\theta [/tex]

    You can figure out how to evaluate this one, as for the second, I have to sleep. If no one has helped you, I'll try to figure it out tomorrow (it looks tough).
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