Post on 23-Dec-2015
7.5Area Between Two Curves
•Find Area Between 2 Curves
•Find Consumer Surplus
•Find Producer Surplus
Area between 2 curvesLet 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) a
b
dx.
Example: Find the area of the region that is bounded by the graphs of
First, look at the graph
of these two functions.
Determine where they
intersect.
(endpoints not given)
f (x) 2x 1 and
.1)( 2 xxg
Example (continued): Second, find the points of intersection by
setting f (x) = g (x) and solving.
f (x) g(x)
2x 1 x2 1
0 x2 2x
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.
(2x 1) (x2 1) 0
2
dx (2x x2 )0
2
dx
x2 x3
3
0
2
22 23
3
02 03
3
4 8
3 0 0 4
3
Example: Find the area bounded by
Answer: 15
2( ) x 1, ( ) 2 4, 1, 2f x g x x x and x
Example: Find the area of the region enclosed by
Answer: 19/3
2 2 , [0,4]y x x and y x on
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
0[ ( ) ]
QD q p dq
Integrate from 0 to the quantityDemand function – price price and quantity are from the equil. pt.
Example: Find the consumer surplus for the demand function given by
When x = 3, we have Then,
D(x) (x 5)2 when x 3.2(3) (3 5) 4.y D
Consumer
Surplus D q p
0
q
(x 5)2 4
0
3
(x 2 10x 21)dx0
3
dq
dq
Example(concluded):
3 2
0
332
0
3 32 2
( 10 21)
5 213
3 05 3 21 3 5 0 21 0
3 3
9 45 63 (0)
$27.00
x x dx
xx x
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
0[ ( )] .qp S q dq
Integrate from 0 to the quantityprice- Supply functionprice and quantity are from the equil. pt.
Example : Find the producer surplus for
Find y when x is 3.
When x = 3, Then,
.3 when 3)( 2 xxxxS
.15333)3( 2 S
32
0
15 ( 3)x x dx Producer Surplus
32
03
2
0
$22.50
(15 3)
( 12)
x x dx
x x dx
Example: Given
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.
D(x) (x 5)2 and S(x) x2 x 3,
Example (continued):a) To find the equilibrium point, set D(x) =
S(x) and solve.
Thus, xE = 2. To find pE, substitute xE into either D(x) or S(x) and solve.
(x 5)2 x2 x 3
x2 10x 25 x2 x 3
10x 25 x 3
22 11x
2 x
Example (continued):If we choose D(x), we have
Thus, the equilibrium point is (2, $9).
pE D xE D 2 2 5 2
3 2
$9
Example (continued):b) The consumer surplus at the equilibrium
point is
(x 5)2dx 290
2
(x 5)3
3
0
2
18
(2 5)3
3
(0 5)3
3 18
27
3
125
3 18
44
3 $14.67
Example (concluded):b) The producer surplus at the equilibrium
point is
29 (x2 x 3)dx 18 x3
3 x
2
2 3x
0
2
0
2
18 (2)3
3
(2)2
2 32
(0)3
3 (0)2
2 30
18 8
3
4
3 6
0
22
3$7.33
More examples:
1)Find the area bounded by
2)Find the area bounded by
3)Given the following functions,
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
, , 1 and 2x xy e y e x x
4 3 2 2( ) 3 4 10, ( ) 40 x over [1,3]f x x x x g x
Demand 50 2000 Supply 10 500p q p q