Find the centroid of the region bonded by the line y=x and the parabola y=x2?

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  • Pope
    Lv 7
    4 weeks ago

    You must mean y = x and y = x², right? Similar problems come up so often that it may be worth your trouble to learn this geometric property of parabolas.

    The region you defined is a parabola segment.

    Its chord joins O(0, 0) and A(1, 1).

    The midpoint of the chord is B(1/2, 1/2).

    The central diameter is the line parallel to the axis and bisecting the chord.

    central diameter: x = 1/2

    The central diameter meets the parabola at C(1/2, 1/4).

    The centroid, K, lies on the central diameter, dividing BC internally in ratio 2:3.

    y-coordinate of K

    = [3(1/2) + 2(1/4)] / (2 + 3)

    = 2/5

    K(1/2, 2/5)

    Attachment image
  • rotchm
    Lv 7
    4 weeks ago

    Hint: First you need to know the concept, or formulas, for the centroid. State them here and we will proceed to show you how to apply them to your problem.

    x_c = ?

    y_c = ?

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