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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


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→∞


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



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


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


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


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] .



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 )(


drrfdttfdxxfb

a

b

a

ba )()()(


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


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

where

n

ii

n

ba xxfdxxf

1

)(lim)(

xiaandn

abx xi


MIDPOINT RULE

n

i

niba xf‧‧‧xfxxxfdxxf

11 )]()([)()(

where

n

abx

and

],1[int)(2

11 iiiii xxofmidpoxxx


dxxfdxxf ba

ab )()(


0)( dxxfaa


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


dxxfdxxfdxxf ba

bc

ca )()()([.5


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


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


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


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

)()()( aFbFdxxfba

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


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 )()(


▓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 )(


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


■ 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.


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

)()()(' aFbFdxxFba


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

dttfxg xa )()(


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.


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


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


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

dxxfab

f baave )(

1


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


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


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


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 )()('))((


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 )()('))(( )(

)(


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


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

dttfx )(0