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find the centroid of the region bounded by x^2-12x and the x-axis

  • calculus -

    The expression x²-12x crosses the x-axis twice, at x=0 and x=12.

    The curve stays below the x-axis on the interval [0,12].


    The area is therefore:
    I1=∫(x^2-12x)dx from x=0 to x=12.
    By symmetry, the centroid lies on the line x=6.
    To find the y-distance, evaluate the integral in which each slice is multiplied by y/2=(x^2-12x), equal to the centroid of each slice:
    The y-position of the centroid is then:

    I get yc=-14.4 (below the x-axis)

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