essential calculus ch04 integrals
DESCRIPTION
ESSENTIAL CALCULUS CH04 Integrals. 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. - PowerPoint PPT PresentationTRANSCRIPT
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