calculus ii chapter 5. definite integral example

CALCULUS II

Chapter 5

Definite Integral

axisx

belowArea

axisx

aboveAreadttf

b

a

)(

Example

4

2

35)(

b

a

dttf

http://www.youtube.com/watch?v=LkdodHMcBuc

Properties of the Definite Integral1:

2:

3:

4:

5:

6:

8:

9:

10:

11:

Indefinite Integrals or Antiderivatives

You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions).

AntiderivativeAn antiderivative of a function f is a function F such that

F f

Ex.

2( ) 3 2F x x

An antiderivative of ( ) 6f x x

since ( ) ( ).F x f x

is

( )f x dxmeans to find the set of all antiderivatives of f.

The expression:

read “the indefinite integral of f with respect to x,”

( )f x dx

Integral sign Integrand

Indefinite Integral

x is called the variable of integration

Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.

Notice 26 3xdx x C

Constant of Integration

Represents every possible antiderivative of 6x.

Power Rule for the Indefinite Integral, Part I

1

if 11

nn xx dx C n

n

Ex.

43

4

xx dx C

Power Rule for the Indefinite Integral, Part II

1 1lnx dx dx x C

x

x xe dx e C Indefinite Integral of ex and bx

ln

xx bb dx C

b

Sum and Difference Rules

( ) ( )kf x dx k f x dx

f g dx fdx gdx Ex.

( constant)k

4 43 32 2 2

4 2

x xx dx x dx C C

2 2x x dx x dx xdx 3 2

3 2

x xC

Constant Multiple Rule

Ex.

http://www.youtube.com/watch?v=Lb8QrUN6Nck

http://www.youtube.com/watch?v=aw_VM_ZDeIo

Table of Indefinite Integrals

http://video.google.com/videoplay?docid=-8662080079069101664#

http://www.youtube.com/watch?v=t3yq21QrGLY&feature=channel

Fundamental Theorem of Calculus (part 1)

• If is continuous for , then tF bta

aFbFdttFb

a


x

a

df t dt f x

dx

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

Visualization


)()(

)()(

],[)(

xfxAdx

d

dttfxA

baoncontinuousxfx

a

The Fundamental Theorem of Calculus

Ex. 3 4 If ( ) 5 , find ( ). x

a

A x t tdt A x

3 4 ( ) 5A x x x

The Fundamental Theorem of Calculus

Ex. dttxA

x

1

23)(

?)3(),2(),0( AAA

?)(xAdx

d?)3(),2(),0( AAA

cos xd

t dtdx cos x 1. Derivative of an integral.



sinxdt

dx

sin sind

xdx

0

sind

xdx

cos x

First Fundamental Theorem:

20

1

1+t

xddt

dx 2

1

1 x

1. Derivative of an integral.



2

0cos

xdt dt

dx

2 2cosd

x xdx

2cos 2x x

22 cosx x

The upper limit of integration does not match the derivative, but we could use the chain rule.

http://www.youtube.com/watch?v=PGmVvIglZx8

53 sin

x

dt t dt

dxThe lower limit of integration is not a constant, but the upper limit is.

53 sin xdt t dt

dx

3 sinx x

We can change the sign of the integral and reverse the limits.

2

2

1

2

x

tx

ddt

dx eNeither limit of integration is a constant.

2 0

0 2

1 1

2 2

x

t tx

ddt dt

dx e e

It does not matter what constant we use!

2 2

0 0

1 1

2 2

x x

t t

ddt dt

dx e e

2 2

1 12 2

22xx

xee

(Limits are reversed.)

(Chain rule is used.)2 2

2 2

22xx

x

ee

We split the integral into two parts.

More Ex’s on the FTC

dttxAx

sin

5

23)( dttxAx

ln

2

3 1)(

dttxAx

7

cos)(

Integration by SubstitutionMethod of integration related to chain rule differentiation. If u is a function of x, then we can use the formula

/

ffdx du

du dx

Integration by Substitution

Ex. Consider the integral: 92 33 5x x dx3 2pick +5, then 3 u x du x dx

10

10

uC

9u du 103 5

10

xC

Sub to get Integrate Back Substitute

23

dudx

x

2Let 5 7 then 10

duu x dx

x

Ex. Evaluate

3/ 21

10 3/ 2

uC

3/ 225 7

15

xC

25 7x x dx

2 1/ 215 7

10x x dx u du

Pick u, compute du

Sub in

Sub in

Integrate

http://www.youtube.com/watch?v=LWp5pWj7s_8&feature=channel

3ln

dx

x xLet ln then u x xdu dx

Ex. Evaluate

3

3ln

dxu du

x x

2

2

uC

2ln

2

xC

http://www.youtube.com/watch?v=WDEZNX7arqY&feature=channel

3

3 2

t

t

e dt

e 3

3Let +2 then

3t

t

duu e dt

e

Ex. Evaluate

3

3

1 1

32

t

t

e dtduue

ln

3

uC

3ln 2

3

teC

Examples

xexxx

xx

x

xexx

x

x

x

sin88

39

16

)3sin()5(

cos4 432

29

4 2

http://www.youtube.com/watch?v=mie4LtRH4q0&feature=channel

http://www.youtube.com/watch?v=xY9Sq0vQ9Bg&feature=channel

Shortcuts: Integrals of Expressions Involving ax + b

Rule

1

1( 1)

nn ax b

ax b dx C na n

1 1lnax b dx ax b Ca

1ax b ax be dx e Ca

1

lnax b ax bc dx c C

a c

Evaluating the Definite Integral

Ex. Calculate

5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 26.39056

http://video.google.com/videoplay?docid=8180465335026223351#

Computing Area Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and

the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0, 2].

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund. Thm. of Calculus

32xy

Examples

2

0

3

1

5.0

0

2

1

2

0

23

1

2

cos

162

dxxedxedxx

dxx

dxxdxx

xx

Substitution for Definite Integrals

Ex. Calculate 1 1/ 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41/ 22 1/ 2

0 02 3x x x dx u du

43/ 2

0

2

3u

16

3

Notice limits change

The Definite Integral As a Total

If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

Total change in quantity ( )b

a

Q r x dx

The Definite Integral As a Total

Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

10

2

Total change in distance ( )v t dt

Net or Total Change as the Integral of a Rate

)()()( tstsdx

dtv

)()()( 12

2

1

tstsdttst

t

Integral of a rate

of change

Total change over

],[ 21 tt

ft

min

minutes

A honey bee makes several trips from the hive to a flower garden.

The velocity graph is shown below.

What is the total distance traveled by the bee?

200ft

200ft

200ft

100ft

200 200 200 100 700 700 feet

ft

min

minutes

What is the displacement of the bee?

200ft

-200ft

200ft

-100ft

200 200 200 100 100 100 feet towards the hive

To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement.

Displacementb

aV t dt

Distance Traveledb

aV t dt

To find distance traveled we have to use absolute value.

Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.)

Or you can use your calculator to integrate the absolute value of the velocity function.

velocity graph

position graph

1

2

1

2

1

2

Displacement:

1 11 2 1

2 2

Distance Traveled:

1 11 2 4

2 2

Examples

• A particle moves along a line so that its velocity at time t is (in meters per second):

• Find the displacement of the particle during the period

• Find the total distance travelled during the same period

6)( 2 tttv

41 t

Examples

• A factory produces bicycles at a rate of

(in t weeks)• How many bicycles were produced from day 8

to 21?

weekperbicyclest95

Examples

• At 7 AM, water begins leaking from a tank at a rate of

(t is the number of hours after 7 AM)• How much water is lost between 9 and 11

AM?

hourgalt /25.02

In the linear motion equation:

dSV t

dt V(t) is a function of time.

For a very small change in time, V(t) can be considered a constant. dS V t dt

S V t t We add up all the small changes in S to get the total distance.

1 2 3S V t V t V t

1 2 3S V V V t

S V t t We add up all the small changes in S to get the total distance.

1 2 3S V t V t V t

1 2 3S V V V t

1

k

nn

S V t

1n

n

S V t

S V t dt

As the number of subintervals becomes infinitely large (and the width becomes infinitely small), we have integration.

This same technique is used in many different real-life problems.

Example 5: National Potato Consumption

The rate of potato consumption for a particular country was:

2.2 1.1tC t

where t is the number of years since 1970 and C is in millions of bushels per year.

For a small , the rate of consumption is constant.t

The amount consumed during that short time is C t t

Example 5: National Potato Consumption

2.2 1.1tC t The amount consumed during that short time is C t t

We add up all these small amounts to get the total consumption:

total consumption C t dt

4

22.2 1.1tdt

4

2

12.2 1.1

ln1.1tt

From the beginning of 1972 to the end of 1973:

7.066million bushels

http://www.youtube.com/watch?v=N2sniUkxxek

http://www.youtube.com/watch?v=EY18ooh3GsA&feature=related

69

Review

• Recall derivatives of inverse trig functions

1

2

12

1

2

1sin , 1

11

tan1

1sec , 1

1

d duu u

dx dxud du

udx u dxd du

u udx dxu u

70

Integrals Using Same Relationships

2 2

2 2

2 2

arcsin

1arctan

1arcsec

du uC

aa udu u

Ca u a adu u

Ca au u a

When given integral problems,

look for these patterns

When given integral problems,

look for these patterns

71

Identifying Patterns

• For each of the integrals below, which inverse trig function is involved?

2

4

13 16

dx

x 225 4

dx

x x

29

dx

x 2 2 10

dx

x x Hint: use completing the square

Hint: use completing the square

72

Warning

• Many integrals look like the inverse trig forms• Which of the following are of the inverse trig

forms?

21

dx

x

21

x dx

x

21

dx

x

21

x dx

x

If they are not, how are they integrated?

If they are not, how are they integrated?

calculus ii chapter 5. definite integral example

Documents

ftc slide

visualization slide

formula slide

definite integral slide

variable of integration

b x slide

indefinite integral

lower limit of integration