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