Find the coordinates of the centroid of the regions bounded by the graphs of:

Question

i)y=x2+1,x=0,x=1 and y=0
ii)y+x2=6 and y+2x-3=0

Answer 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