7.3 area and the definite integral and 7.4 the fundamental theorem of calculus objectives evaluate...

7.3 Area and The Definite Integraland

7.4 The Fundamental Theorem of Calculus

OBJECTIVES Evaluate a definite integral. Find the area under a curve over a given

closed interval. Interpret an area below the horizontal axis. Solve applied problems involving definite

integrals.

Fundamental theorem of calculus:for a continuous function f on the

interval[a, b]

where F is any anti-derivative of f. a and b are called the lower and upper

limits of integration.

( ) ( ) ( ) ( )b b

aaf x dx F x F b F a

The notation F(b) - F(a) means to evaluate the anti-derivative at b and

subtract the anti-derivative evaluated at a. Since both F(b) and

F(a) contain the constant of integration c, they will cancel each

other out, thus eliminating c altogether.

Evaluate each definite integral

6

2

0

8 17 30x x dx

Evaluate each definite integral

83

0

12xdx

100.04

0

12.2956xe dx

Evaluate each definite integral

Example: Evaluate each of the following:

4 2

1

3

0

1

a) ( ) ;

b) ;

1c) 1 2 (assume 0).

x

e

x x dx

e dx

x dx xx

Example (continued):

a) (x2 x) 1

4

dx x3

3x2

2

1

4

43

3

42

2

( 1)3

3

( 1)2

2

64

3

16

2

1

3

1

2

64

3 8

1

3

1

214

1

6

Example (continued):

b) ex dx0

3

ex 0

3 e3 e0

e3 1

Example (continued):

c) 1 2x 1

x

1

e

dx x x2 ln x 1

e

(e e2 lne) (112 ln1)

(e e2 1) (11 0)

e e2 1 1 1

e e2 3

The definite integral denoted

is defined to be the area of the region between the curve of f(x) and the x-axis bounded by the vertical lines at a and b. IF

b

adxxf )(

( ) 0f x

( )b

af x dx

For example the definite integral

the area of the region between the graph of x^2 and the x-axis from x = 1 to x = 2. See next slide for graph.

2

1

2dxx

2)( xxf

Example: Suppose that y is the profit per mile traveled (in thousands of dollars) and x is number of miles traveled, in thousands. Find the area under y = 1/x over the interval [1, 4] and interpret the significance of this area.

1.3863 thousand dollars

The area represents a total profit of $1386.30 when the miles traveled increase from 1000 to 4000 miles.

4 4

11

1ln

ln 4 ln1

ln 4 0

1.3863

dx xx

To find area when for the values of x in the interval [a, b], find the absolute value of the definite integral.

When finding area below the x-axis from [a, b], if you just find the definite integral, you will get a negative answer. Area can’t be negative, so just take the absolute value of the definite integral.

( ) 0f x

Example: Find the area under the curve of from x = -1 to x = 1.

Hint: First graph the function to see if f(x) >0.

2( ) 8f x x

Example: Predict the sign of the integral by using what you know about area.

Consider ( x3 3x 1)dx 1

2

.

From the graph, it appears that there is considerably more area below the x-axis than above. Thus, we expect that the sign of the integral will be negative.

This is worked on the next slide.

Evaluating the integral not the area, we have

2 3

1

242

1

4422

3 1

3

4 2

12 3 32 2 1 1

4 2 4 2

1 34 6 2 1

4 2

10 2

41

24

x x dx

xx x

Determine the area under the given curve for the values of x.

4 x to1 x from 32 xy

A)Find the area under the curve of over the interval [6, 10].

*Remember to look at the graph first. Is f(x) above or below the x-axis?

B) Find the area under the curve of from x = 0 to x = 5.

( ) 2 14f x x

2( ) 2 2f x x

Example: Northeast Airlines determines that the marginal profit resulting from the sale of x seats on a jet traveling from Atlanta to Kansas City, in hundreds of dollars, is given by

Find the total profit when 60 seats are sold.

P (x) x 6.

Slide 4.3 - 25 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example (continued): We integrate

60

0

60

0

603 2

0

3 2 3 2

6

26

3

2 260 6 60 0 6 0

3 3

50.1613

P x dx

x dx

x x

Example (concluded): When 60 seats are sold, Northeast’s profit is –$5016.13. That is, the airline will lose $5016.13 on the flight.

Given the marginal cost and a fixed cost, the total cost for producing the first n units is found by

0

'( ) fixed costnC x dx

A company has a marginal cost given by

0.02'( ) 6 xC x eAnd fixed cost of $500.(a) find the total cost for producing the first 100 units.(b) find the additional cost if production is raised from 100 to 150 units.

Given marginal revenue, then the total revenue of raising the production level from a units to b units is found by

dxxRb

a )('

The marginal revenue for an item is R’(x) = 25 - 0.02x. Find the change in revenue if sales are increased from 100 to 160 units.

Given marginal profit, the additional profit obtained from raising the production level from a units to b units is found by

'( )

b

aP x dx

A company has a marginal profit function given by P’(x) = 165 – 0.1x. (a)Find the change in profit if production is increased from 1500 to 2000 units.(b) Find the profit for the 1500th unit.

ExampleThe Harris Company found that its rate of profit (in thousands of dollars) after t years of operation is given by

a) Find the total profit in the first three years. (to the nearest whole number)b) Find the profit in the fourth year of operation. (to the nearest whole number)c) How would you set up the profit for years 2 through 6?

12 3'( ) (3 3)(t 2 2)P t t t