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Math Help - Coordinates of a centroid

  1. #1
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    Coordinates of a centroid

    The coordinates of the centroid for f(x) = x "for all" epislon[0,2] are (4/3, 2/3); how far away is the centroid for the planner figure governed by g(x) = 3x^2 along the same interval

    a. (3/2, 18/5)
    b. Exactly 3.9 units
    c. 2.9 units
    d. 1.5 units
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by shanniepooh2 View Post
    The coordinates of the centroid for f(x) = x "for all" epislon[0,2] are (4/3, 2/3); how far away is the centroid for the planner figure governed by g(x) = 3x^2 along the same interval

    a. (3/2, 18/5)
    b. Exactly 3.9 units
    c. 2.9 units
    d. 1.5 units
    The x coordinate of the centroid of g(x)=3x^2 \ \forall \ \varepsilon[0,2] is:

    \bar x=\frac{1}{\int_0^2 3x^2\,dx}\int_0^2\int_0^{3x^2}x\,dy\,dx=\frac{1}{8  }\int_0^2\left.\bigg[xy\bigg]\right|_{y=0}^{3x^2}\,dx

    =\frac{1}{8}\int_0^2 3x^3\,dx=\frac{3}{32}\left.\bigg[x^4\bigg]\right|_0^2=\color{red}\boxed{\frac{3}{2}}

    From how I interpreted the problem, D appears to be the answer...

    Does this make sense?

    --Chris
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