5.1. Area Between Two Curves http://www.ck12.org
A=
∫b
a[f(x)−g(x)]dx.Example 1:
Find the area of the region enclosed betweeny=x^2 andy=x+ 6.
Figure 2
Solution:
We first make a sketch of the region (Figure 2) and find the end points of the region. To do so, we simply equate the
two functions,
x^2 =x+ 6,
and then solve forx.
x^2 −x− 6 = 0
(x+ 2 )(x− 3 ) = 0from which we getx=−2 andx=3.
So the upper and lower boundaries intersect at points(− 2 , 4 )and( 3 , 9 ).
As you can see from the graph,x+ 6 ≥x^2 and hencef(x) =x+6 andg(x) =x^2 in the interval[− 2 , 3 ]. Applying the
area formula,