1 / 17

# 7.5 Area Between Two Curves - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' 7.5 Area Between Two Curves' - eugenia-garrett

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 7.5Area Between Two Curves

Find Area Between 2 Curves

Find Consumer Surplus

Find Producer Surplus

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

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)

Second, find the points of intersection by setting f (x) = g(x) and solving.

2

ò

ò

é

ù

+

-

+

=

-

2

2

(

2

1

)

(

1

)

(

2

)

x

x

d

x

x

x

d

x

ë

û

0

0

2

é

ù

3

x

=

-

2

x

ê

ú

3

ë

û

0

æ

ö

æ

ö

3

3

2

0

=

-

-

-

2

2

2

0

ç

÷

ç

÷

è

ø

è

ø

3

3

8

4

=

-

-

+

=

4

0

0

3

3

Example (concluded):

Lastly, compute the integral. Note that on [0, 2], f (x) is the upper graph.

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.

Suppose that p = D(x) describes the demand function for a commodity. Then, the consumer surplus is defined for the point (Q, P) as

o

n

s

u

m

e

r

x

(

)

ò

=

E

D

x

d

x

x

p

-

×

E

E

S

u

r

p

l

u

s

0

3

ò

=

×

2

(

5

)

3

4

x

d

x

-

-

0

3

ò

=

2

-

(

1

0

2

5

)

1

2

x

x

d

x

+

-

0

Example: Find the consumer surplus for the demand function given by

When x = 3, we have Then,

Suppose that p = S(x) is the supply function for a commodity. Then, the producer surplus is defined for the point (Q, P) as

r

o

d

u

c

e

r

x

ò

(

)

E

x

p

S

x

d

x

=

×

-

E

E

S

u

r

p

l

u

s

0

3

ò

×

-

2

3

1

5

(

3

)

x

x

d

x

=

+

+

0

Example : Find the producer surplus for

Find y when x is 3.

When x = 3, Then,

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.

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.

If we choose D(x), we have

Thus, the equilibrium point is (2, \$9).

b) The consumer surplus at the equilibrium point is

é

ù

3

2

x

x

2

ò

2

×

-

+

+

2

9

(

3

)

1

8

3

x

x

d

x

x

-

+

+

=

ê

ú

3

2

ë

û

0

0

æ

ö

æ

ö

3

2

3

2

(

2

)

(

2

)

(

0

)

(

0

)

=

×

+

+

×

1

8

3

2

3

0

-

+

+

-

ç

÷

ç

÷

è

ø

è

ø

3

2

3

2

æ

ö

8

4

2

2

=

1

8

6

0

-

+

+

-

=

ç

÷

è

ø

3

3

3

»

\$

7

.

3

3

Example (concluded):

b) The producer surplus at the equilibrium point is