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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

    [itex]x^{2/3}+y^{2/3}=2^{2/3}[/itex]

    [itex]\frac{x^2}{9}+\frac{y^2}{4}=1[/itex]

    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:

    [itex]A_1=\int_0^2(y_1-y_2)dx[/itex]

    To find the area of plane 2:

    [itex]A_2=\int_2^3y_1dx[/itex]

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

    [itex]M_{1y}=\int_0^2x(y_1-y_2)dx[/itex]

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

    [itex]M_{1x}=\frac{1}{2}\int_0^2(y_1+y_2)(y_1-y_2)dx[/itex]

    And similiarly,

    [itex]M_{2y}=\int_2^3xy_1dx[/itex]
    [itex]M_{2x}=\frac{1}{2}\int_2^3y_1^2dx[/itex]

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

    [itex]\int_0^2\sqrt{4-\frac{4}{9}x^2}dx-\int_0^2\left({2^{2/3}-x^{2/3}}\right)^{3/2}dx[/itex]

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

    [itex]3\arcsin{\frac{2}{3}}+\frac{2\sqrt{5}}{3}[/itex]

    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

    AKG

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    Science Advisor
    Homework Helper

    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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