7.5
Area Between Two Curves
•Find Area Between 2 Curves
•Find Consumer Surplus
•Find Producer Surplus
Area between 2 curves
Let f and g be continuous functions and suppose
that f (x) ≥ g (x) over the interval [a, b]. Then the
area of the region between the two curves, from
x = a to x = b, is
  f (x)  g(x) dx.
b
a
Example: Find the area of the region that
is bounded by the graphs of
f (x)  2x  1 and
2
g ( x)  x  1.
First, look at the graph
of these two functions.
Determine where they
intersect.
(endpoints not given)
Example (continued):
Second, find the points of intersection by
setting f (x) = g (x) and solving.
f (x)

2x  1 
0

g(x)
x 1
2
x  2x
2
0
 x(x  2)
x  0 or x  2
Example (concluded):
Lastly, compute the integral. Note that on
[0, 2], f (x) is the upper graph.

2
0
(2x  1)  (x  1)  dx 
2


2
0
(2x  x 2 ) dx
2
 2 x 
x  
3 0

3
 2 23   2 03 
 2   0  

3 
3

8
4 00 
3
4
3
Example: Find the area bounded by
f ( x)   x  1, g ( x)  2 x  4, x  1, and x  2
2
Answer: 15
Example: Find the area of the region enclosed by
y  x 2  2 x, and y  x on [0, 4]
Answer: 19/3
DEFINITION:
The equilibrium point, (xE, pE), is the point at
which the supply and demand curves
intersect.
It is that point at which
sellers and buyers come
together and purchases
and sales actually occur.
DEFINITION:
Suppose that p = D(x) describes the demand
function for a commodity. Then, the
consumer surplus is defined for the point
(Q, P) as
Integrate from 0 to the quantity
Demand function – price
price and quantity are from the equil. pt.

Q
0
[ D(q)  p]dq
Example: Find the consumer surplus for
the demand function given by
D(x)  (x  5)2 when x  3.
2
y

D
(3)

(3

5)
 4.
When x = 3, we have
Then,
Consumer
Surplus



0



3
0
D (q )  p dq
q
3
2
(x  5)  4
0
dq
2
( x  10 x  21)dx
Example(concluded):
3
  ( x 2  10 x  21)dx
0
3
x

2
   5 x  21x 
3
0
3
3
 33



0
2
2
   5  3  21  3     5  0  21  0 
3
 3

 ( 9  45  63)  (0)
 $27.00
DEFINITION:
Suppose that p = S(x) is the supply function
for a commodity. Then, the producer
surplus is defined for the point (Q, P) as
Integrate from 0 to the quantity
price- Supply function
price and quantity are from the equil. pt.

q
0
[ p  S (q)]dq.
Example : Find the producer surplus for
S ( x)  x 2  x  3 when x  3.
Find y when x is 3.
2
S
(
3
)

3
 3  3  15.
When x = 3,
Producer Surplus

Then,
3
2  x  3)dx
15

(
x

0
3
2
(15

x

0
3
2
(

x


0
$22.50

x  3)dx
x  12)dx
Example: Given
2
2
D(x)  (x  5) and S(x)  x  x  3,
find each of the following:
a) The equilibrium point.
b) The consumer surplus at the
equilibrium point.
c) The producer surplus at the equilibrium
point.
Example (continued):
a) To find the equilibrium point, set D(x) =
S(x) and solve.
(x  5)2
x 2  10x  25
10x  25
22
2





x2  x  3
x2  x  3
x3
11x
x
Thus, xE = 2. To find pE, substitute xE into
either D(x) or S(x) and solve.
Example (continued):
If we choose D(x), we have
pE  D (xE ) 
D (2 )

(2  5 )
2
(3)


2
$9
Thus, the equilibrium point is (2, $9).
Example (continued):
b) The consumer surplus at the equilibrium
point is

2
0
(x  5) dx  2  9 
2



2
 (x  5) 
 3   18

0
3
(2  5)3 (0  5)3

 18
3
3
27 125
44
 
 18 
3
3
3
$14.67
Example (concluded):
b) The producer surplus at the equilibrium
point is
2
x

x
2  9   (x  x  3)dx  18     3x 
0
3 2
0
2
3
2
2
 (2)3 (2)2
  (0)3 (0)2

 18  

 3  0

 3  2  
 3
  3

2
2
8 4

22
 18     6   0 
3 3

3
 $7.33
More examples:
1) Find the area bounded by
y  e x , y  e x , x  1 and x  2
2) Find the area bounded by
f ( x)  x 4  3x3  4 x 2  10, g ( x)  40  x 2 over [1,3]
3) Given the following functions,
Demand p  50q  2000
Supply p  10q  500
Find
a) the Equilibrium Point
b) Producer Surplus
c) Consumer Surplus
Answers: 1) 6.611 2) 488/5 or 97.6 3) a) (25, $750), b) $3125, c) $15,625