Chapter 5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Chapter 5

The Definite Integral


5.1

Estimating with Finite Sums


Quick Review Solutions


What you’ll learn about

Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output

… and why

Learning about estimating with finite sums sets the

foundation for understanding integral calculus.


Example Finding Distance Traveled when Velocity Varies

2A particle starts at 0 and moves along the -axis with velocity ( )

for time 0. Where is the particle at 3?

x x v t t

t t

Graph and partition the time interval into subintervals of length . If you use

1/ 4, you will have 12 subintervals. The area of each rectangle approximates

the distance traveled over the subint

v t

t

erval. Adding all of the areas (distances)

gives an approximation to the total area under the curve (total distance traveled)

from 0 to 3.t t


Example Finding Distance Traveled when Velocity Varies

2

Continuing in this manner, derive the area 1/ 4 for each subinterval and

add them:

1 9 25 49 81 121 169 225 289 361 441 529 2300

256 256 256 256 256 256 256 256 256 256 256 256 2568.98

im


LRAM, MRAM, and RRAM approximations to the area under the graph of y=x2 from x=0 to x=3


Example Estimating Area Under the Graph of a Nonnegative Function

2Estimate the area under the graph of ( ) sin from 0 to 3.f x x x x x

Apply the RAM program (found in the that

accompanies this textbook).

Using 1000 subintervals, you find the left endpoint

approximate area of 5.77476

Technology Resource Manual


5.2

Definite Integrals


Quick Review

42

1

4

1

3 3 3

42

1

Evaluate the sum.

1.

2. 3 1

Write the sum in sigma notation.

3. 2 3 4 ... 49 50

4. 2 4 6 8 ... 98 100

5. 3(1) 3(2) ... 3(100)

6. Write the expression as a single sum in sigma notation

n

k

n

n

k

n

4

1

0

0

3

7. Find 1 if is odd.

8. Find 1 if is even.

n

n k

k

n k

k

n

n

n



50

4

2

50

1

1003

1

2

1

4

1

3 3 3

Evaluate the sum.

1.

2. 3 1

Write the sum in sigma notation.

3. 2 3 4 ... 49 50

4. 2 4 6 8 ... 98 100

5. 3(1) 3(2) ... 3(100)

6. Writ

30

34

2

e the expres

3

k

k

k

n

k

n

k

k

k

k

4 42

1 1

0

0

42

1sion as a single sum in sigma notation 3

7. Find 1 if is odd.

8. Find 1 if is even.

3

0

1

nn n

n k

k

n k

k

n n

n

n n

n



Riemann Sums The Definite Integral Computing Definite Integrals on a Calculator Integrability

… and whyThe definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus.


Sigma Notation

1 2 3 11

...n

k n nk

a a a a a a


The Definite Integral as a Limit of Riemann Sums

-1

0

Let be a function defined on a closed interval [ , ]. For any partition

of [ , ], let the numbers be chosen arbitrarily in the subinterval [ , ].

If there exists a number such that lim

k k k

P

f a b P

a b c x x

I 1

( )

no matter how and the 's are chosen, then is on [ , ] and

is the of over [ , ].

n

k kk

k

f c x I

P c f a b

I f a b

integrable

definite integral


The Existence of Definite Integrals

All continuous functions are integrable. That is, if a function is

continuous on an interval [ , ], then its definite integral over

[ , ] exists.

f

a b

a b


The Definite Integral of a Continuous Function on [a,b]

1

Let be continuous on [ , ], and let [ , ] be partitioned into subintervals

of equal length ( - ) / . Then the definite integral of over [ , ] is

given by lim ( ) , where each is chon

k kn k

f a b a b n

x b a n f a b

f c x c

th

sen arbitrarily in the

subinterval.k


The Definite Integral

( )b

a f x dx


Example Using the Notation

th

2

1

The interval [-2,4] is partitioned into subintervals of equal length 6 / .

Let denote the midpoint of the subinterval. Express the limit

lim 3 2 5 as an integral.

k

n

k kn k

n x n

m k

m m x

2 4 2

21

lim 3 2 5 3 2 5n

k kn km m x x x dx


Area Under a Curve (as a Definite Integral)

If ( ) is nonnegative and integrable over a closed interval [ , ],

then the area under the curve ( ) from to is the

, ( ) .b

a

y f x a b

y f x a b

A f x dx

integral

of from to f a b


Area

Area= ( ) when ( ) 0.

( ) area above the -axis area below the -axis .

b

a

b

a

f x dx f x

f x dx x x


The Integral of a Constant

If ( ) , where is a constant, on the interval [ , ], then

( ) ( ) b b

a a

f x c c a b

f x dx cdx c b a


Example Using NINT

2

-1Evaluate numerically. sinx xdx

NINT( sin , , -1,2) 2.04x x x


5.3

Definite Integrals and Antiderivatives


Quick Review

1

Find / .

1. -sin

2. cos

3. ln(sec )

4. ln(cos )

5. ln

6.

7. tan

x

dy dx

y x

y x

y x

y x

y x x

y xe

y x



1

2

Find / .

1. -sin

2. cos

3. ln(sec )

4. ln

/ cos

/ sin

/ tan

/ t(cos )

5. ln

6.

7. tan

an

/ 1 ln

/

1/

1

x xx

dy dx

y x

y x

y x

y x

y

dy dx x

dy dx x

dy dx x

dy dx x

dy dx x

dy dx xe e

x x

y xe

dy dxx

y x



Properties of Definite Integrals Average Value of a Function Mean Value Theorem for Definite Integrals Connecting Differential and Integral Calculus

… and whyWorking with the properties of definite integrals helpsus to understand better the definite integral. Connectingderivatives and definite integrals sets the stage for theFundamental Theorem of Calculus.


Rules for Definite Integrals


Example Using the Rules for Definite Integrals

1 4 1

-1 1 -1

1

4

Suppose ( ) 5, ( ) 2, and ( ) 7 .

Find ( ) if possible.

f x dx f x dx h x dx

f x dx

1

4 ( ) 2f x dx



1 4 1

-1 1 -1

4

1

Suppose ( ) 5, ( ) 2, and ( ) 7 .



f x dx

4

1 ( ) 5 ( 2) 3f x dx



1 4 1

-1 1 -1

2

2

Suppose ( ) 5, ( ) 2, and ( ) 7 .



h x dx

Not enough information is given.


Average (Mean) Value

If is integrable on [ , ], its average (mean) value on [ , ] is

1( ) ( )b

a

f a b a b

avg f f x dxb a


Example Applying the Definition

2Find the average value of ( ) 2 on [0,4].f x x

4 2

0

1( ) ( )

1 2 Use NINT to evaluate the integral.

4 01 40

4 3

10

3

b

aavg f f x dxb a

x dx


The Mean Value Theorem for Definite Integrals

If is continuous on [ , ], then at some point in [ , ],

1( ) ( ) .b

a

f a b c a b

f c f x dxb a


The Mean Value Theorem for Definite Integrals


The Derivative of an Integral

( ) ( ).x

a

df t dt f x

dx


Quick Quiz Sections 5.1 - 5.3

You should solve the following problems without using a calculator.

1. If ( ) 2 , then ( ) 3

(A) 2 3

(B) 3 -3

(C) 4 -

(D) 5 - 2

(E) 5 -3

b b

a af x dx a b f x dx

a b

b a

a b

b a

b a



You should solve the following problems without using a calculator.

1. If ( ) 2 , then

(D)

( ) 3

(A) 2 3

(B) 3 -3

(C) 4 -

(E)

5 - 2

5 -3

b b

a af x

b

dx a b f x dx

a b

b a

a

b a

a

b



1

0

1

0

1

0

1

0

20

0

1 1 2 3 202. The expression ...

20 20 20 20 20

is a Riemann sum approximation for

(A) 20

(B)

1(C)

20 201

(D) 201

(E) 20

xdx

xdx

xdx

xdx

xdx



1

0

1

0

1

0

20

1

0

0

1 1 2 3 202. The expression ...

20 20 20 20 20

is a Riemann sum approxima

(B)

tion for

(A) 20

1(C)

20 201

(D) 201

(E) 20

xdx

xdx

x

dx

x

dx

x

x

d



2

23. What are all values of for which 0?

(A) -2

(B) 0

(C) 2

(D) -2 and 2

(E) -2, 0, and 2

kk x dx



2

23. What are all values of for which 0?

(A) -2

(B) 0

(D) -2 and 2

(E) -2, 0, and 2

(C) 2

kk x dx


5.4

Fundamental Theorem of Calculus


Quick Review

3

3

2 2

Find / .

1. sin

2. sin

3. ln 3- ln 7

4. sin cos

5. 3

6. cos

7. sin and 2

8. / 2

x

dy dx

y x

y x

y

y x x

y

xy

xy t x t

dx dy x



3

3

2 2

2 3

2

2

Find / .

1. sin

2. sin

3. ln 3- ln 7

4. s

/ 3 cos

/ 3 sin cos

/ 0

/ 0

/ 3 ln 3

cos si

in cos

5. 3

6. cos

7. sin a

n/

cos

nd

x x

dy dx

y x

y x

y

y x x

y

dy dx x x

dy dx x x

dy dx

dy dx

dy dx

x x xdy

xy

x

y t

x

x

dx

2

8. / 2

cos/

21

/2

t

dx d

tdy dx

dy dxx

y x



Fundamental Theorem, Part 1 Graphing the Function Fundamental Theorem, Part 2 Area Connection Analyzing Antiderivatives Graphically

… and why

The Fundamental Theorem of Calculus is a Triumph of

Mathematical Discovery and the key to solving many

problems.

( )x

a f t dt


The Fundamental Theorem of Calculus

If is continuous on [ , ], then the function ( ) ( )

has a derivative at every point in [ , ], and

( ) ( ).

x

a

x

a

f a b F x f t dt

x a b

dF df t dt f x

dt dx


The Fundamental Theorem of Calculus

( ) ( )

Every continuous function is the derivative of some other function.

Every continuous function has an antiderivative.

The processes of integration and differentiation are inverses of o

x

a

df t dt f x

dx

f

ne another.


Example Applying the Fundamental Theorem

Find sin .xdtdt

dx

sin sinxdtdt x

dx


Example The Fundamental Theorem with the Chain Rule

2

1Find / if sin .xdy dx y tdt

2

1

2

1

1

sin

sin and .

Apply the chain rule:

sin

sin

x

u

u

y tdt

y tdt u x

dy dy du

dx du dxd du

tdtdu dx

duu

dx

2

sin 2

2 sin

u x

x x


Example Variable Lower Limits of Integration

5Find if sin .x

dyy t tdt

dx

5

5

5

sin sin

sin

sin

x

x

x

d dt tdt t tdt

dx dxd

t tdtdxx x


The Fundamental Theorem of Calculus, Part 2

If is continuous at every point of [ , ], and if is any antiderivative

of on [ , ], then ( ) ( ) - ( ).

This part of the Fundamental Theorem is also called the

.

b

a

f a b F

f a b f x dx F b F a Integral

Evaluation Theorem


The Fundamental Theorem of Calculus, Part 2

( ) ( ) ( )

Any definite integral of any continuous function can be calculated without

taking limits, without calculating Riemann sums, and often without effort -

so long as an antiderivative

b

a f x dx F b F a

f

of can be found.f


Example Evaluating an Integral

3 2

-1Evaluate 3 1 using an antiderivative.x dx

33 2 3

-1 1

33

3 1

3 3 1 1

32

x dx x x


How to Find Total Area Analytically

To find the area between the graph of ( ) and the -axis over the interval

[ , ] analytically,

1. partition [ , ] with the zeros of ,

2. integrate over each subinterval,

3. add the absolute values o

y f x x

a b

a b f

f

f the integrals.


How to Find Total Area Numerically

To find the area between the graph of ( ) and the -axis over the

interval [ , ] numerically, evaluate

NINT(| ( ) |, , , )

y f x x

a b

f x x a b


5.5

Trapezoidal Rule


Quick Review

4 2

3

Tell whether the curve is concave up or concave down on the given interval.

1. cos on [-1,0]

2. 3 6 on [8,17]

3. sin on [48 ,50 ]2

4. on [-5,5]

5. 1/ on [4, 8]

6. csc

x

y x

y x x

xy

y e

y x

y x

on 0,

7. sin - cos on [1,2]y x x



4 2

3

concav

Tell whet

e dow

her the curve is concave up or concave down on the given interval.

1. cos on [-1,0]

2. 3 6 on [8,17]

3. sin on [48

n

conca

,50

ve up

concave do] 2

4.

wn

y x

y x x

xy

y e

on [-5,5]

5. 1/ on [4, 8]

6. c

concave up

concave up

concasc on 0,

7. sin - cos on [1,2

ve up

conca] ve d own

x

y x

y x

y x x



Trapezoidal Approximations Other Algorithms Error Analysis

… and whySome definite integrals are best found bynumerical approximations, and rectangles are notalways the most efficient figures to use.


Trapezoidal Approximations

0 1 11 2

0

1 2 1

0 1 2 1

0 1 1 1 1

( ) ...2 2 2

...2 2

2 2 ... 2 ,2

where ( ), ( ), ..., ( ), ( ).

b n n

a

n

n

n n

n n n

y y y yy yf x dx h h h

y yh y y y

hy y y y y

y f a y f x y f x y f b


The Trapezoidal Rule

0 1 2 1

To approximate ( ) , use

2 2 ... 2 ,2

where [ , ] is partitioned into n subintervals of equal length

( - ) / .

LRAM RRAMEquivalently, ,

2where LRAM and RRAM are the Rienamm

b

a

n n

n n

n n

f x dx

hT y y y y y

a b

h b a n

T

sums using the left

and right endpoints, respectively, for for the partition.f


Simpson’s Rule

0 1 2 3 2 1

To approximate ( ) , use

4 2 4 ... 2 4 ,3

where [ , ] is partitioned into an even number subintervals

of equal length ( - ) / .

b

a

n n n

f x dx

hS y y y y y y y

a b n

h b a n


Error Bounds

( 4 )

2 4

If and represent the approximations to ( ) given by the

Trapezoidal Rule and Simpson's Rule, respectively, then the errors

and satisfy

and 12 180

n

b

a

T s

T sf f

T S f x dx

E E

b a b aE h M E h M


Quick Quiz Sections 5.4 and 5.5

You may use a graphing calculator to solve the following problems

1. The function is continuous on the closed interval [1,7] and has

values that are given below:

f

x 1 4 6 7

f(x) 10 30 40 20

7

1

Using the subintervals [1,4], [4,6], and [6,7], what is the trapezoidal

approximation of ( ) ?

(A) 110

(B) 130

(C) 160

(D) 190

(E) 210

f x dx




You may use a graphing calculator to solve the following problems

1. The function is continuous on the closed interval [1,7] and has

values that are given below:

f

x 1 4 6 7

f(x) 10 30 40 20

7

1

Using the subintervals [1,4], [4,6], and [6,7], what is the trapezoidal

approximation of ( ) ?

(A)

(C) 160

110

(B) 130

(D) 190

(E) 210

f x dx



32. Let ( ) be an antiderivative of sin . If (1) 0, then (8)

(A) 0.00

(B) 0.021

(C) 0.373

(D) 0.632

(E) 0.968

F x x F F



32. Let ( ) be an antiderivative of sin . If (1) 0, then (8)

(A) 0.00

(B) 0.0

(

21

(C) 0.373

D) 0.632

(E) 0.968

F x x F F



2 23

-23. Let ( ) . At what value of is ( ) a minimum?

(A) For no value of

(B) 1/2

(C) 3/2

(D) 2

(E) 3

x x tf x e dt x f x

x



2 23

-23. Let ( ) . At what value of is ( ) a minimum?

(A) For no value of

(B) 1/2

(C)

(D) 2

(E

3/2

) 3

x x tf x e dt x f x

x


Chapter Test

3

4

Let be the region in the first quadrant enclosed by the -axis and the graph

of the function 4 - .

1. Sketch the rectangles and compute by hand the area for the MRAM

approximations.

2. Sketch the t

R x

y x x

4rapeziods and compute by hand the area for the T

approximations.


Chapter Test

2 5 5

-2 2 -2

2

5

5

-2

3. Suppose ( ) 4, ( ) 3, ( ) 2.

Which of the following statements are true, and which, if any, are false?

(a) ( ) 3

(b) ( ) ( ) 9

(c) ( ) ( ) on the interval -

f x dx f x dx g x dx

f x dx

f x g x dx

f x g x

1 3 2

0

/ 2 2

0

2

0

2 5

4. Find the total area between the curve and the -axis given 4 - , 0 6.

Evaluate using the Integral Evaluation Theorem.

5. 8 12 5

6. sec

27. Evaluate:

1

x

x y x x

s s ds

d

dyy


Chapter Test

8. A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced.

(a) Give an upper estimate and a lower estimate for the amount of oil consumed by the generator during that week.

(b) Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.


Chapter Test

3

2

3

0

9. Find / . 2 cos

10. Solve for : 2 3 4

x

x

dy dx y tdt

x t t dt

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