4.3 Copyright © 2014 Pearson Education, Inc. Area and Definite Integrals OBJECTIVE Find the area under a curve over a given closed interval. Evaluate a

4.3

Copyright © 2014 Pearson Education, Inc.

Area and Definite Integrals

OBJECTIVE• Find the area under a curve over a given closed interval.

• Evaluate a definite integral.• Interpret an area below the horizontal axis.• Solve applied problems involving definite integrals.

Slide 4- 2Copyright © 2014 Pearson Education, Inc.

To find the area under the graph of a nonnegative, continuous function f over the interval [a, b]:

1. Find any antiderivative F(x) of f (x). (The simplest is the one for which the constant of integration is 0.)

2. Evaluate F(x) using b and a, and compute F(b) – F(a). The result is the area under the graph over the interval [a, b].

4.3 Area and Definite Integrals


Example 1: Find the area under the graph of y = x2 +1 over the interval [– 1, 2].

1. Find any antiderivative F(x) of f (x). We choose the simplest one.

F(x) x3

3 x



Example 1 (concluded):2. Substitute 2 and –1, and find the difference

F(2) – F(– 1).


6

8 1

2 13 3

8 1

2 13 3

(2) ( 1)F F

3 32 ( 1)2 ( 1)

3 3



Quick Check 1

Refer to the function in Example 1.

a.) Calculate the area over the interval

b.) Calculate the area over the interval

c.) Can you suggest a shortcut for part (b)?

0,5 .

2,2 .

a.) 3

2 1, so .3

xf x x F x x

Substitute 0 and 5, and find the difference 5 0 .F F

5 0F F125

53

2

463

3 35 0

5 03 3



Quick Check 1 Concluded

b.) Calculate the area over the interval [ 2,2].

2 2F F

33 222 2

3 3

2 24 4

3 3

19

3

c.) Can you suggest a shortcut for part (b)?

Thus we would integrate from 0 to 2, then double the results.

Note that Then we have 2 2 .F F

2 2 2 2 2 2 .F F F F F


DEFINITION:

Let f be any continuous function over the interval [a, b] and F be any antiderivative of f. Then, the definite integral of f from a to b is

f (x) dxa

b

F(b) F(a).



Example 2: Evaluate

Using the antiderivative F(x) = x3/3, we have

It is convenient to use an intermediate notation:

where F(x) is an antiderivative of f (x).

x2dxa

b

.

x2dxa

b

b3

3

a3

3.

( ) ( ) ( ) ( ),

b b

aaf x dx F x F b F a



Example 3: Evaluate each of the following:


1

1c) 1 2 (assume 0).

ex dx x

x

3

0b) ;xe dx

4 2

1a) ( ) ;x x dx


Example 3 (continued):


4 2

1a) ( )x x dx

43 2

13 2

x x

3 2 3 24 4 ( 1) ( 1)

3 2 3 2

64 16 1 1

3 2 3 2

64 1 1 1

8 143 3 2 6


Example 3 (continued):


3 1e

3 0e e3

0b) xe dx

3

0

xe


Example 3 (concluded):


2 3e e

2 1 1 1e e

2( 1) (1 1 0)e e

2 2( ln ) (1 1 ln1)e e e

2

1ln

ex x x

1

1c) 1 2

ex dx

x



Quick Check 2

Evaluate each of the following:

a.)

b.)

c.)

4 3

22 3x x dx

ln 4

02 xe dx

5

1

1xdx

x



Quick Check 2 Continued

a.) 4 3

22 3x x dx

44 2

2

1 3

2 2x x

4 2 4 21 3 1 34 4 2 2

2 2 2 2

128 24 8 6

102



ln 4

02 xe dx

Quick Check 2 Continued

b.) ln 4

02 xe ln 4 02 2e e 2 4 2 1 6

c.)5

1

1xdx

x

5

1

11 dx

x 5

1lnx x 5 ln5 1 ln1

5 ln5 1 0 4 ln5 2.39


THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS

If a continuous function f has an antiderivative F over [a, b], then

limn

f (xi )xi1

n

f (x) da

b

x F(b) F(a).



Example 4: Suppose that y is the profit per mile traveled 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

ln 4 0

ln 4 ln1

4

1ln x

4

1

dx

x


Example 4 (concluded): The area represents a total profit of $1386.30 when the miles traveled increase from 1000 to 4000 miles.



Example 5: Predict the sign of the integral by examining the graph, and then evaluate the integral.

From the graph, it appears that there is considerablymore area below the x-axisthan above. Thus, we expectthat the sign of the integralwill be negative.

Consider ( x3 3x 1)dx 1

2

.



Example 5 (concluded): Evaluating the integral, we have


1 1

0 2 24 4

1 34 6 2 1

4 2

4422 12 3 3

2 2 1 14 2 4 2

242 3 2

11

33 1

4 2

xx x dx x x


Example 6: 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.



Example 6 (continued): We integrate to find P(60).


60

060P P x dx

50.1613

3 2 3 22 260 6 60 0 6 0

3 3

603 2

0

26

3x x

60

06x dx


Example 6 (concluded): When 60 seats are sold, Northeast’s profit is –$5016.13. That is, the airline will lose $5016.13 on the flight.




Quick Check 3

Referring to Example 6, find the total profit of Northeast Airlines when 140 seats are sold.

From Example 6, we have

140

0140 6P x dx

1403/2

0

26

3x x

3/2 3/22 2140 6 140 0 6 0

3 3

264.3349

When 140 seats are sold, Northeast Airlines makes$100 264.3349 $26,433.49.


Example 7: A particle starts out from some origin. Its velocity, in miles per hour, is given by

where t is the number of hours since the particle left the origin. How far does the particle travel during the second, third, and fourth hours (from t = 1 to t = 4)?

v(t) t t,



Example 7 (continued):Recall that velocity, or speed, is the rate of change of distance with respect to time. In other words, velocity is the derivative of the distance function, and the distance function is an antiderivative of the velocity function. To find the total distance traveled from t = 1 to t = 4, we evaluate the integral

4

1 .t t dt



Example 7 (concluded):


1

12 mi.6

16 16 2 1 14 15 73

3 2 3 2 3 2 6

3 2 2 3 2 22 1 2 14 4 1 1

3 2 3 2

43 2 2

1

2 1

3 2t t

4 4 1 2

1 1t t dt t t dt



Section Summary

• The exact area between the x-axis and the graph of the nonnegative continuous function over the interval is found by evaluating the definite integral

where F is an antiderivative of f.

• If a function has areas both below and above the x-axis, the definite integral gives the net total area, or the difference between the sum of the areas above the x-axis and the sum of the areas below the x-axis.

y f x ,a b

,b

af x dx F b F a



Section Summary Concluded

If there is more area above the x-axis than below, the definite integral will be positive.

If there is more area below the x-axis than above, the definite integral will be negative.

If the areas above and below the x-axis are the same, the definite integral will be 0.

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4.3 Copyright © 2014 Pearson Education, Inc. Area and Definite Integrals OBJECTIVE Find the area under a curve over a given closed interval. Evaluate a