math

Find the coordinates of the centroid of the regions bounded by the graphs of:
i)y=x2+1,x=0,x=1 and y=0
ii)y+x2=6 and y+2x-3=0

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  1. I'll show you how to do one of them.

    i) For the x coordinate of the centroid, integrate
    (Integral of) x(x^2+1) dx from x=0 to x=1
    = [x^4/4 +x^2/2]@x=1 - [x^4/4 -x^2/2]@x=0
    = 1/4 + 1/2 = 3/4
    For the y coordinate of the centroid, integrate
    (Integral of) (y/2)(x^2+1) dx from x=0 to x=1
    = (Integral of) (1/2)(x^2+1)^2 dx from x=0 to x=1
    = (Integral of) (1/2)(x^4+2x^2+1)^2 dx from x=0 to x=1
    = (1/2)[x^5/5 +(2/3)x^3 +x] @ x=1
    - (1/2)[x^5/5 +(2/3)x^3 +x] @ x=0
    = 1/10 + 1/3 + 1/2 = 14/15

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