Posted by James on Monday, June 24, 2013 at 9:18am.
The area of the region is
A = ∫[-1,3] (2x+3)-x^2 dx
= -1/3 x^3 + x^2 + 3x [-1,3]
= (-9+9+9)-(1/3 + 1 - 3)
= 32/3
For xbar, find
X = ∫[-1,3] x((2x+3)-x^2) dx
= -1/4 x^4 + 2/3 x^3 + 3/2 x^2 [-1,3]
= (-81/4 + 18 + 27/2)-(-1/4 + 2/3 + 3/2)
= 32/3
For ybar, find
Y = ∫[-1,3] 1/2 ((2x+3)^2 - (x^2)^2) dx
= -1/5 x^5 + 4/3 x^3 + 6x^2 + 9x [-1,3]
= 544/15
xbar = X/A = 1
ybar = Y/A = 17/5
To solve this category of problems where you need the area/centroid of a region bounded by two curves, you need to first find TWO intersection points of the two curves by equating y1(x)=x^2 and y2(x)=2*x+3.
The intersection points are at x=-1 and x=3, with y2(x) above y1(x).
Then you need to find the area by integrating
A=∫(y2(x)-y1(x))dx between the limits x=-1 and x=3.
=32/3
To find the centroid(xBar,yBar), you need to find the first moments,
xBar=∫x(y2(x)-y1(x))dx / A
and
yBar=∫(1/2)(y2(x)+y1(x))(y2(x)-y1(x))dx /A
from which I get
(xBar,yBar)=(1,17/5)