Area of a Region Between Two
Curves
Derek Nelson
6/8/2005
Definition
-
=
(Area of region between f and g)
(Area of region under f)
b
b
A=∫ a [f(x) – g(x)]dx
A=∫ a f(x)dx
(Area of region under g)
b
A=∫ a g(x)dx
If f and g are continuous on [a,b] and g(x) < f(x) for all x in [a,b], then the area of
the region bounded by the graphs of f and g and the vertical lines x = a and x = b is
A=∫ [f(x) – g(x)]dx
b
a
Proof
After partitioning the interval [a,b] into n subintervals, each of width ∆x, and sketch a representative rectangle
of width ∆x and height f(xi) – g(xi), where xi is the ith interval, as shown in Figure 6.3. The area of this
representative rectangle is
∆Ai = (height)(width) = [f(xi) – g(xi)] ∆x
By adding the areas of the n rectangles and taking the limit as ||∆||→0 (n→∞)
n
Σ
lim
[f(xi] – g(xi)] ∆x
n→∞ i=1
Since f and g are continuous on [a,b], f-g is also continuous on this interval and the limit exists. Therefore, the
area A of the given region is
i=1
A = lim
a
Σ
[f(xi] – g(xi)] ∆x
n→∞
A=∫b [f(x) – g(x)]dx
Example 1
Find the area of the region bounded by the graphs of y = x2 + 2, y = -x, x= 0, and x = 1
A   x 2  2   x dx
1
0
A   x 2  x  2dx
1
0
1
x x

A     2 x
3 2
0
3
2
1 1

A     2   0
3 2 
17
A  units 2
6
Example 2
Find the area of the region bounded by the graphs of f(x) = 2-x2 and g(x) = x
A
A
1

2  x   xdx
2
2
1

2  x
2
2
2 x  x
x2  x  2  0
 x  2 x  1  0
x  2, 1
1
x x 

A  2 x   
3 2  2

3
2
 x dx
2
8 
 1 1 
A   2       4   2
3 
 3 2 
9
A  units 2
2
Example 3
Find the area of the region bounded by the graphs of f(x) = 3x3-x2-10x and g(x) = -x2+2x