ESSENTIAL CALCULUSESSENTIAL CALCULUS

CH04 IntegralsCH04 Integrals

In this Chapter:In this Chapter:

4.1 Areas and Distances

4.2 The Definite Integral

4.3 Evaluating Definite Integrals

4.4 The Fundamental Theorem of Calculus

4.5 The Substitution Rule

Review

Chapter 4, 4.1, P194

Chapter 4, 4.1, P195

Chapter 4, 4.1, P195

Chapter 4, 4.1, P195

Chapter 4, 4.1, P195

Chapter 4, 4.1, P195

Chapter 4, 4.1, P196

Chapter 4, 4.1, P197

Chapter 4, 4.1, P197

Chapter 4, 4.1, P198

Chapter 4, 4.1, P198

Chapter 4, 4.1, P199

Chapter 4, 4.1, P199

2. DEFINITION The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:

A=lim Rn=lim[f(x1)∆x+f(x2) ∆x+‧‧‧+f(xn) ∆x] n→∞ n→∞

Chapter 4, 4.1, P199

Chapter 4, 4.1, P199

This tells us toend with i=n.

This tells usto add.

This tells us tostart with i=m.

xxf i

n

mi

)(

xxf‧‧‧xxfxxfxxf ni

n

i

)()()()( 21

1

Chapter 4, 4.1, P199

Chapter 4, 4.1, P200

The area of A of the region S under the graphs of the continuous function f is

A=lim[f(x1)∆x+f(x2) ∆x+‧‧‧+f(xn) ∆x]

A=lim[f(x0)∆x+f(x1) ∆x+‧‧‧+f(xn-1) ∆x]

A=lim[f(x*1)∆x+f(x*2) ∆x+‧‧‧+f(x*n) ∆x]

n→∞

xxfc

n

cn

)(lim

1

xxf c

n

cn

)(lim 1

1

n→∞

n→∞

xxfc

n

cn

)*(lim

1

Chapter 4, 4.2, P205

FIGURE 1 A partition of [a,b] with sample points *ix

Chapter 4, 4.2, P205

A Riemann sum associated with a partition P and a function f is constructed by evaluating f at the sample points, multiplying by the lengths of the corresponding subintervals, and adding:

ni

n

ixxf‧‧‧xxfxxfxxf

ni

)()()()( *

2*

1*1

*

1 2

Chapter 4, 4.2, P206

FIGURE 2A Riemann sum is the sum of theareas of the rectangles above thex-axis and the negatives of the areasof the rectangles below the x-axis.

2. DEFINITION OF A DEFINITE INTEGRAL If f is a function defined on [a,b] ,the definite integral of f from a to b is the number

n

iii

x

ba xxfdxxf

1

*

0max)(lim)(

1

provided that this limit exists. If it does exist, we say that f is integrable on [a,b] .

Chapter 4, 4.2, P206

Chapter 4, 4.2, P206

NOTE 1 The symbol ∫was introduced by Leibniz and is called an integral sign. Itis an elongated S and was chosen because an integral is a limit of sums. In the notation is called the integrand and a and b are called the limits of integration;a is the lower limit and b is the upper limit. The symbol dx has no official meaning by itself; is all one symbol. The procedure of calculating an integralis called integration.

)(,)( xfdxxfba

dxxfba )(

Chapter 4, 4.2, P206

drrfdttfdxxfb

a

b

a

ba )()()(

Chapter 4, 4.2, P207

3. THEOREM If f is continuous on [a,b], or if f has only a finite number of jump discontinuities, then f is integrable on [a,b]; that is, the definite integral dx exists. )(xfba

Chapter 4, 4.2, P207

4. THEOREM If f is integrable on [a,b], then

where

n

ii

n

ba xxfdxxf

1

)(lim)(

xiaandn

abx xi

Chapter 4, 4.2, P208

Chapter 4, 4.2, P208

Chapter 4, 4.2, P208

Chapter 4, 4.2, P208

Chapter 4, 4.2, P210

Chapter 4, 4.2, P211

Chapter 4, 4.2, P211

MIDPOINT RULE

n

i

niba xf‧‧‧xfxxxfdxxf

11 )]()([)()(

where

n

abx

and

],1[int)(2

11 iiiii xxofmidpoxxx

Chapter 4, 4.2, P212

dxxfdxxf ba

ab )()(

Chapter 4, 4.2, P212

0)( dxxfaa

Chapter 4, 4.2, P213

Chapter 4, 4.2, P213

Chapter 4, 4.2, P213

PROPERTIES OF THE INTEGRAL Suppose all the following integrals exist.

where c is any constant

where c is any constant

),(.1 abccdxba

dxxgdxxfdxxgxf ba

ba

ba )()()]()([.2

,)()(.3 dxxfcdxxcf ba

ba

dxxgdxxfdxxgxf ba

ba

ba )()()]()([.4

Chapter 4, 4.2, P214

Chapter 4, 4.2, P214

dxxfdxxfdxxf ba

bc

ca )()()([.5

Chapter 4, 4.2, P214

COMPARISON PROPERTIES OF THE INTEGRAL

6. If f(x)≥0 fpr a≤x≤b. then

7.If f(x) ≥g(x) for a≤x≤b, then

8.If m ≤f(x) ≤M for a≤x≤b, then

.0)( dxxfba

.)()( dxxgdxxf ba

ba

)()()( abMdxxfabm ba

Chapter 4, 4.2, P215

Chapter 4, 4.3, P217

29.The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.

(a) (b)

(c) (d)

dxxf )(20 dxxf )(5

0

dxxf )(75 dxxf )(9

0

Chapter 4, 4.3, P217

30. The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral.(a) (b) (c)dxxg )(2

0 dxxg )(62 dxxg )(7

0

Chapter 4, 4.3, P218

EVALUATION THEOREM If f is continuous on the interval [a,b] , then

)()()( aFbFdxxfba

Where F is any antiderivative of f, that is, F’=f.

Chapter 4, 4.3, P220

the notation ∫f(x)dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus

The connection between them is given by the Evaluation Theorem: If f is continuous on [a,b], then

baba dxxfdxxf )()(

Chapter 4, 4.3, P220

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

dxxfba )(dxxf )(

Chapter 4, 4.3, P220

1. TABLE OF INDEFINITE INTEGRALS

dxxfcdxxcf )()( dxxgdxxfdxxgxf )()()]()([

Ckxkdx )1(1

1

ncn

xdxfx

nn

Cxxdx cossin Cxxdx sincos

Cxxdx tansec2

Cxxdxx sectansec

Cxxdx cotcsc2

Cxdxx csccotcsc

Chapter 4, 4.3, P221

■ Figure 3 shows the graph of the integrandin Example 5. We know from Section 4.2 that the value of the integral can be interpreted as the sum of the areas labeled with a plus sign minus the area labeled with a minus sign.

Chapter 4, 4.3, P222

NET CHANGE THEOREM The integral of a rate of change is the net change:

)()()(' aFbFdxxFba

Chapter 4, 4.4, P227

The Fundamental Theorem deals with functions defined by an equation of the from

dttfxg xa )()(

Chapter 4, 4.4, P227

Chapter 4, 4.4, P227

Chapter 4, 4.4, P227

Chapter 4, 4.4, P227

Chapter 4, 4.4, P229

Chapter 4, 4.4, P229

THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1 If f is continuous on [a,b] , then the function defined by

dttfxg xa )()( a≤x≤b

is an antiderivative of f, that is, g’(x)=f(x) for a<x<b.

Chapter 4, 4.4, P231

THE FUNDAMENTAL THEOREM OF CALCULUS Suppose f is continuous on [a,b].

1. If g(x)= f(t)dt, then g’(x)=f(x).2. f(x)dx=F(b)-F(a), where F is any antiderivative of f,

that is, F’=f.

xa

ba

Chapter 4, 4.4, P231

We noted that Part 1 can be rewritten as

which says that if f is integrated and the result is then differentiated, we arrive backat the original function f.

)()( xfdttfdx

d xa

Chapter 4, 4.4, P232

we define the average value of f on the interval [a,b] as

dxxfab

f baave )(

1

Chapter 4, 4.4, P233

THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on [a,b], then there exists a number c in [a,b] such that

dxxfab

fcf baave )(

1)(

that is,

))(()( abcfdxxfba

Chapter 4, 4.4, P234

1.Let g(x)= , where f is the function whose graph is shown.(a) Evaluate g(0),g(1), g(2) ,g(3) , and g(6).(b) On what interval is g increasing?(c) Where does g have a maximum value?(d) Sketch a rough graph of g.

dttfx )(0

Chapter 4, 4.4, P234

2.Let g(x)= , where f is the function whose graph is shown.(a) Evaluate g(x) for x=0,1,2,3,4,5, and 6.(b) Estimate g(7).(c) Where does g have a maximum value? Where does it have a minimum value?(d) Sketch a rough graph of g.

dttfx )(0

Chapter 4, 4.4, P235

Chapter 4, 4.4, P235

Chapter 4, 4.5, P237

4. THE SUBSTITUTION RULE If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

duufdxxgxgf )()('))((

Chapter 4, 4.5, P239

5.THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS If g’ is continuous on [a,b] and f is continuous on the range of u=g(x), then

duufdxxgxgf bgag

ba )()('))(( )(

)(

Chapter 4, 4.5, P240

6. INTEGRALS OF SYMMETRIC FUNCTIONS Suppose f is continuous on [-a,a].

(a)If f is even [f(-x)=f(x)], then

(b)If f is odd [f(-x)=-f(x)], then

.)(2)( 0 dxxfdxxf aaa

.0)( dxxfaa

Chapter 4, 4.5, P240

Chapter 4, 4.5, P240

Chapter 4, 4.5, P245

5. The following figure shows the graphs of f, f’, and . Identify each graph, and explain your choices.

dttfx )(0