The Definite Integral

(1)

Slide 5- 1

(2)

Chapter 5

(3)

5.1

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Slide 5- 4

Quick Review

2

1. A train travels at 80 mph for 4 hours. How far does it travel? 2. Beginning at a standstill, a car maintains a constant acceleration of 5 ft/sec for 10 seconds. What is its velocity after 10 seconds? 3. A pump working at 100 gallons/minute pumps for two hours. How many gallons are pumped?

4. At 8:00pm, the temperature began dropping at a rate of 1 degree Celcius per hour. Twelve hours later it began rising at a rate of 1.5 degrees per hour for six hours. What was the net change in

(5)

Slide 5- 5

Quick Review Solutions

2

1. A train travels at 80 mph for 4 hours. How far does it travel? 2. Beginning at a standstill, a car maintains a constant acceleration of 5 ft/sec for 10 seconds. What is its velo

320 mi

cit

les

y after 10 seconds? 3. A pump working at 100 gallons/minute pumps for two hours.

How many gallons are pumped?

4. At 8:00pm, the temperature began dropping at a ra

50 ft/s

12,000 g

te of 1 degree Ce

allons

lcius per hour. Twelve hours later it began rising at a rate of 1.5 degrees per hour for six hours. What was the net change in

(6)

Slide 5- 6

What you’ll learn about



Distance Traveled



Rectangular Approximation Method (RAM)



Volume of a Sphere



Cardiac Output

… and why

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Slide 5- 7

Example

Finding Distance Traveled when

Velocity Varies

2

A 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

  

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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

256 256 256 256 256 256 256 256 256 256 256 256 256

8.98

i m

           

(9)

Slide 5- 9

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Slide 5- 10

Example

Estimating Area Under the

Graph of a Nonnegative Function

2

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

Apply the RAM program (found in the that

accompanies this textbook).

Using 1000 subintervals, you find the left endpoint approximate area of 5.77476

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5.2

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Slide 5- 12

Quick Review





4 2 1 4 1

3 3 3

4 2 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.

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Quick Review Solutions





50 4 2 50 1 100 3 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 4 2 1 1 0 0 4 2 1 sion as a single sum in sigma notation 3

7. Find 1 if is odd.

8. Find 1 if is even.

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just as the derivative is the basis of differential

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Slide 5- 15

Sigma Notation

1 2 3 1

1 ...

n

k n n

k a a a a a  a

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Slide 5- 16

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 k k

k

f c x I

P c f a b

I f a b



  

(17)

Slide 5- 17

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

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Slide 5- 18

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 n ( ) , where each is cho_k _k

n _k

f a b a b n

x b a n f a b

f c x c

 _

  

th

sen arbitrarily in the subinterval.

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Slide 5- 19

The Definite Integral

( )

b

a f x dx

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Slide 5- 20

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 k

n _k

n x n

m k

m m x

 _

 

  

 



2



₄



₂



2 1

lim n 3 _k 2 _k 5 3 2 5

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Slide 5- 21

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



 

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Slide 5- 22

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



 

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Slide 5- 23

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

 



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Slide 5- 24

Example

Using NINT

2

-1

Evaluate numerically. _ xsin xdx

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5.3

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Slide 5- 26

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 x

y xe

y  x

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Slide 5- 27

Quick Review Solutions

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 x x dy dx y x y x y x y x y

dy dx x

dy dx xe e

x x y xe

dy dx x

y  x

(28)

Slide 5- 28

What you’ll learn about



Slide 5- 30

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 f x dx( ) 2

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Slide 5- 31

Example

Using the Rules for Definite

Integrals

1 4 1

-1 1 -1

4 1

f x dx



  



   

4

1 f x dx( ) 5 ( 2) 3



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Slide 5- 32

Example

Using the Rules for Definite

Integrals

1 4 1

-1 1 -1

2 2

h x dx



  



   

(33)

Slide 5- 33

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 dx

b a 

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Slide 5- 34

Example

Applying the Definition

2

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





4 2 0 1 ( ) ( ) 1

2 Use NINT to evaluate the integral. 4 0 1 40 4 3 10 3 b a

avg f f x dx

(35)

Slide 5- 35

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 dx

b a 

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Slide 5- 36

(37)

Slide 5- 37

The Derivative of an Integral

( ) ( ).

x a d

f t dt f x

(38)

Slide 5- 38

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 f x dx a b a f x dx

a b

b a

a b

b a

     

(39)

Slide 5- 39

Quick Quiz

Sections 5.1 - 5.3





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 f x a

b

dx a b f x dx

a b

b a

Slide 5- 45

Quick Review





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

y x x

y

x y

x

y t x t

dx dy x

(46)

Slide 5- 46

Quick Review Solutions









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

/ 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

dy dx dy dx

x x x

dy x y x y t x x dx                2

8. / 2

(47)

… 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

(48)

Slide 5- 48

The Fundamental Theorem of Calculus

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

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

( ) ( ).

x a

f a b F x f t dt

x a b

dF d

f t dt f x dt dx





(49)

Slide 5- 49

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 d

f t dt f x dx

f

 

(50)

Slide 5- 50

Example

Applying the Fundamental

Theorem

Find d xsin .

tdt dx 

sin sin

x d

tdt x

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Slide 5- 51

Example

The Fundamental Theorem

with the Chain Rule

2

1

Find / if x sin .

dy 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 dx

(52)

Slide 5- 52

Example

Variable Lower Limits of

Integration

5

Find if x sin .

dy

y t tdt

dx  









5

sin sin

sin

x x

x

d d

t tdt t tdt

dx dx

d

t tdt dx

x x

 



 

(53)

Slide 5- 53

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

(54)

Slide 5- 54

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

  

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Slide 5- 55

Example

Evaluating an Integral





3 2 -1

Evaluate 3_ x 1 dx using an antiderivative.











    





3

3 2 3

-1 ₁

3 3

3 1

3 3 1 1 32

x dx x x _

   

     

(56)

Slide 5- 56

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



(57)

Slide 5- 57

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

(58)

5.5

(59)

Slide 5- 59

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

x y y e y x y x          _{ }    

 on 0,





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



(60)

Slide 5- 60

Quick Review Solutions

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

x y y e          _{ }   





on [-5,5] 5. 1/ on [4, 8] 6. c

concave up concave up

conca sc on 0,

7. sin - cos on [1,2

ve up conca

] ve down x

y x

y x x



(61)

Slide 5- 61

What you’ll learn about



Trapezoidal Approximations



Other Algorithms



Error Analysis

… and why

Some definite integrals are best found by

(62)

Slide 5- 62

Trapezoidal Approximations





0 1 1 2 1

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

y y y y y y

f x dx h h h

y y

h y y y

h

y y y y y

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

(63)

Slide 5- 63

The Trapezoidal Rule



0 1 2 1



To approximate ( ) , use

2 2 ... 2 ,

2

where [ , ] is partitioned into n subintervals of equal length ( - ) / .

LRAM RRAM

Equivalently, ,

2

where LRAM and RRAM are the Rienamm

b a

n n

f x dx h

T y y y y y

a b

h b a n

T           

(64)

Slide 5- 64

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 h

S y y y y y y y

a b n

h b a n

 



       

(65)

Slide 5- 65

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 n 180

b a

T s

T f s f

T S f x dx

E E

b a b a

E h M E h M



 

(66)

Slide 5- 66

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

(67)

Slide 5- 67

Quick Quiz

Sections 5.4 and 5.5

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)

(C) 160

110 (B) 130

(D) 190 (E) 210

f x dx

(68)

Slide 5- 68

Quick Quiz

Sections 5.4 and 5.5

3

2. 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

(69)

Slide 5- 69

Quick Quiz

Sections 5.4 and 5.5

3

2. 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

(70)

Slide 5- 70

Quick Quiz

Sections 5.4 and 5.5

2 ₃ 2

-2

3. 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 t

f x e dt x f x

x





(71)

Slide 5- 71

Quick Quiz

Sections 5.4 and 5.5

2 ₃ 2

-2

3. 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 t

f x e dt x f x

x





(72)

Slide 5- 72

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

4

(73)

Slide 5- 73

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

2 7. Evaluate:

1

x

x y x x

s s ds

(74)

Slide 5- 74

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.

(75)

Slide 5- 75

Chapter Test





3 2

3 0

9. Find / . 2 cos

10. Solve for : 2 3 4

x

dy dx y tdt

x t t dt



 

(76)

Slide 5- 76

Chapter Test Solutions

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

4.125

.

2. Sketch

R x

y  x x

4

the trapeziods and compute by hand the area for the T approximations.

(77)

Slide 5- 77

Chapter Test Solutions

3

4

Let R be the region in the first quadrant enclosed by the x-axis and the graph of the function y 4x – x .

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

4.125

.

2. Sketch the trapeziods and compute by hand the area for the T4

approximations.

(78)

Slide 5- 78

Chapter Test Solutions





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 Tru

(

e Tr

b) ( ) ( ) 9 (c) ( ) (

ue

f x dx f x dx g x dx

f x dx

f x g x dx f x g

            





1 3 2 0

/ 2 2 0

) on the interval - 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

Fal

5 6. sec

se

10

3

x x

x y x x

s s ds d              2 0 2

7. Evaluate: 2 l 3

1dy n

y

(79)

Slide 5- 79

Chapter Test Solutions

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. Upper = 4.392 L; Lower = 4.008 L

(80)

Slide 5- 80

Chapter Test Solutions





3 2

3 0

3

9. Find / . 2 cos

10. Solve for :

2 cos

1.63052 or -3.09131

2 3 4

x

dy dx y tdt

x t t dt x x x



 

  

