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 yaxis 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!