Chapter 6The Integral

Sections 6.1, 6.2, and 6.3

The Indefinite Integral

Substitution

The Definite Integral As a Sum

The Definite Integral As Area

The Integral

A physicist who knows the velocity of a particle might wish to know its position at a given time.

A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time.

Introduction

In each case, the problem is to find a function F whose derivative is a known function f.

If such a function F exists, it is called an antiderivative of f.

Antiderivatives

Definition

A function F is called an antiderivative of f on

an interval I if F’(x) = f (x) for all x in I.

For instance, let f (x) = x2.

• It is not difficult to discover an antiderivative of f if we keep the Power Rule in mind.

• In fact, if F(x) = ⅓ x3, then F’(x) = x2 = f (x).

Antiderivatives

However, the function G(x) = ⅓ x3 + 100 also satisfies G’(x) = x2.

• Therefore, both F and G are antiderivatives of f.

Indeed, any function of the form H(x)=⅓ x3 + C, where C is a constant, is an antiderivative of f.

• The question arises: Are there any others?

Antiderivatives

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is

F(x) + C

where C is an arbitrary constant.

Theorem

Antiderivatives

Going back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.

Antiderivatives

Family of Functions By assigning specific values to C, we obtain a

family of functions.

• Their graphs are vertical

translates of one another.

• This makes sense, as each

curve must have the same

slope at any given value

of x.

Notation for Antiderivatives

The symbol is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f .

Thus, means F’(x) = f (x)

( )f x dx

( ) ( )F x f x dx

( )f x dxThe expression:

read “the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f.

( )f x dx

Integral sign Integrand

x is called the variable of integration

Indefinite Integral

For example, we can write

• Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).

3 32 2because

3 3

x d xx dx C C x

dx

Indefinite Integral

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

Example: 26 3xdx x C

Represents every possible antiderivative of 6x.

Constant of Integration

1

if 11

nn x

x dx C nn

Example:4

3

4

xx dx C

Power Rule for the Indefinite Integral

1 1lnx dx dx x C

x

x xe dx e C

Indefinite Integral of ex and bx

ln

xx b

b dx Cb

Power Rule for the Indefinite Integral

Sum and Difference Rules

f g dx fdx gdx

Example:

2 2x x dx x dx xdx 3 2

3 2

x xC

( ) ( )kf x dx k f x dx ( constant)k

4 43 32 2 2

4 2

x xx dx x dx C C

Constant Multiple Rule

Example:

Example - Different Variable

Find the indefinite integral:

273 2 6ue u du

u

213 7 2 6ue du du u du du

u

323 7ln 6

3ue u u u C

Position, Velocity, and Acceleration Derivative Form

If s = s(t) is the position function of an object at time t, then

Velocity = v = Acceleration = a = ds

dtdv

dt

Integral Form

( ) ( )s t v t dt ( ) ( )v t a t dt

Integration by Substitution

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

/

ff dx du

du dx

Example: 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

Integration by Substitution

2Let 5 7 then 10

duu x dx

x

Example: 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

3ln

dx

x xLet ln then u x xdu dx

3

3ln

dxu du

x x

2

2

uC

2ln

2

xC

Example: Evaluate

3

3 2

t

t

e dt

e 3

3Let +2 then

3t

t

duu e dt

e

3

3

1 1

32

t

t

e dtdu

ue

ln

3

uC

3ln 2

3

teC

Example: Evaluate

Let f be a continuous function on [a, b]. If F is any antiderivative of f defined on [a, b], then the definite integral of f from a to b is defined by

( ) ( ) ( )b

af x dx F b F a

The Definite Integral

( )b

af x dx is read “the integral, from a to b of f (x) dx.”

In the notation ,

f (x) is called the integrand.

a and b are called the limits of integration; a is the lower limit and b is the upper limit.

For now, the symbol dx has no meaning by itself; is all one symbol. The dx simply indicates that the independent variable is x.

( )b

af x dx

Notation

The procedure of calculating an integral is called

integration. The definite integral is a

number. It does not depend on x.

Also note that the variable x is a “dummy variable.”

( )b

af x dx

( ) ( ) ( )b b b

a a af x dx f t dt f r dr

The Definite Integral

Geometric Interpretationof the Definite Integral

The Definite Integral As Area

The Definite Integral As Net Change of Area

If f is a positive function defined for a ≤ x ≤ b,

then the definite integral represents the

area under the curve y = f (x) from a to b

( )b

af x dx

( )b

aA f x dx

Definite Integral As Area

If f is a negative function for a ≤ x ≤ b, then the

area between the curve y = f (x) and the x-axis

from a to b, is the negative of ( ) .b

af x dx

Definite Integral As Area

Area from to ( )b

aa b f x dx

Consider y = f (x) = 0.5x + 6 on the interval [2,6]

whose graph is given below,

Definite Integral As Area

6

2Find ( )

) by using geometry

) by using the definition

of definite integral

f x dx

a

b

Definite Integral As Area

6

2( ) Area of

Trapezoid

f x dx

Consider y = f (x) = 0.5x + 6 on the interval [2,6]

whose graph is given below,

( )b

af x dx Area of R1 – Area of R2 + Area of R3

a b

R1

R2

R3

If f changes sign on the interval a ≤ x ≤ b, then definite integral represents the net area, that is, a difference of areas as indicated below:

Definite Integral as Net Area

a b

R1

R2

R3

If f changes sign on the interval a ≤ x ≤ b, and we need to find the total area between the graph and the x-axis from a to b, then

Total Area

c d

Total Area Area of R1 + Area of R2 + Area of R3

Area of R1 ( )a

cf x dx

Area of R2 ( )d

cf x dx

Area of R3 ( )d

bf x dx

Example: Use geometry to compute the integral

5

1

1x dx

Area = 2

5

1

1 8 2 6x dx

Area = 8

Area Using Geometry

( ) 1y f x x

–1

5

Example: Use an antiderivative to compute the integral

5

1

1x dx

Area Using Antiderivatives

First, we need an antiderivative of ( ) 1y f x x

21( ) 1 . Thus,

2F x x dx x x C

5

1

15 3

2 21 (5) ( 1) 6C Cx dx F F

Example: Now find the total area bounded by the curve and the x-axis from x –1 to x 5.

Area Using Antiderivatives

( ) 1y f x x

( ) 1y f x x

–11 5

R1

R2

Total Area Area of R1 + Area of R2

( ) 1y f x x

–11 5

R1

R2

Area of R1

121

11

( ) 22

xf x dx x

Area of R2

525

11

( ) 82

xf x dx x

Total Area 2 + 8 10

Evaluating the Definite Integral

Example: Calculate5

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

Substitution for Definite Integrals

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

Example: Calculate

Computing Area Example: 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

22 .y x

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

Example: 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

The Definite Integral As a Total