Find the centroid for an area defined by the equations: y=2x+4 and y=((2x-3)^2)+1?

Need to use integration

thank you

1 Answer

  • First, find the points of intersection. By equating the values of y:

    2x + 4 = (2x – 3)^2 + 1.

    This solves to x = 1/2 and x = 3, so the limits of integration are from 1/2 to 3. Then, since 2x + 4 > (2x – 3)^2 + 1, we see that the upper function is 2x + 4 and the lower function is (2x – 3)^2 + 1.

    The moments about the x and y-axis are:

    (a) M(x) = p/2 ∫ {(2x + 4)^2 – [(2x – 3)^2 + 1]^2} dx (from x=1/2 to 3) = 625p/12

    (b) M(y) = p ∫ x{2x + 4 – [(2x – 3)^2 + 1]} dx (from x=1/2 to 3) = 875p/48.

    (p is the density of the region, which will cancel out when we find the centroid.)

    Since the mass of the region is:

    A = p ∫ {2x + 4 – [(2x – 3)^2 + 1]} dx (from x=1/2 to 3) = 125p/12,

    the centroid is located at:

    (M(y)/M, M(x)/M)

    = ((875p/48)/(125p/12), (625p/12)/(125p/12))

    = (7/4, 5).

    I hope this helps!

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