626
Chapter 6 | Applications of Integration
Figure 6.4 A region between two curves is shown where one curve is always greater than the other.
We have
a b
A = ∫
⎡ ⎣ f ( x )− g ( x ) ⎤
⎦ dx
4 ⎡ ⎣ ( x +4)−
4 ⎡
⎤ ⎦ dx = ∫
⎤ ⎦ dx
⎛ ⎝ 3− x 2
⎞ ⎠
= ∫
⎣ 3 x
2 +1
1
1
⎦ | 1 4
⎡ ⎣ 3 x 2
x ⎤
⎛ ⎝ 16− 7 4
⎞ ⎠ = 57 4 .
=
=
4 +
The area of the region is 57
2 .
4 units
If R is the region bounded by the graphs of the functions f ( x ) = x 2 +5
6.1
and g ( x ) = x + 1 2
over the
interval ⎡
⎤ ⎦ , find the area of region R .
⎣ 1, 5
In Example 6.1 , we defined the interval of interest as part of the problem statement. Quite often, though, we want to define our interval of interest based on where the graphs of the two functions intersect. This is illustrated in the following example. Example 6.2 Finding the Area of a Region between Two Curves 2 If R is the region bounded above by the graph of the function f ( x ) =9−( x /2) 2 and below by the graph of the function g ( x ) =6− x , find the area of region R .
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online