ESSENTIAL CALCULUS CH04 Integrals. In this Chapter: 4.1 Areas and Distances 4.2 The Definite...
-
Upload
clare-black -
Category
Documents
-
view
215 -
download
0
Transcript of ESSENTIAL CALCULUS CH04 Integrals. In this Chapter: 4.1 Areas and Distances 4.2 The Definite...
![Page 1: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/1.jpg)
ESSENTIAL CALCULUSESSENTIAL CALCULUS
CH04 IntegralsCH04 Integrals
![Page 2: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/2.jpg)
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
![Page 3: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/3.jpg)
Chapter 4, 4.1, P194
![Page 4: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/4.jpg)
Chapter 4, 4.1, P195
![Page 5: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/5.jpg)
Chapter 4, 4.1, P195
![Page 6: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/6.jpg)
Chapter 4, 4.1, P195
![Page 7: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/7.jpg)
Chapter 4, 4.1, P195
![Page 8: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/8.jpg)
Chapter 4, 4.1, P195
![Page 9: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/9.jpg)
Chapter 4, 4.1, P196
![Page 10: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/10.jpg)
Chapter 4, 4.1, P197
![Page 11: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/11.jpg)
Chapter 4, 4.1, P197
![Page 12: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/12.jpg)
Chapter 4, 4.1, P198
![Page 13: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/13.jpg)
Chapter 4, 4.1, P198
![Page 14: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/14.jpg)
Chapter 4, 4.1, P199
![Page 15: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/15.jpg)
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→∞
![Page 16: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/16.jpg)
Chapter 4, 4.1, P199
![Page 17: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/17.jpg)
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
)(
![Page 18: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/18.jpg)
xxf‧‧‧xxfxxfxxf ni
n
i
)()()()( 21
1
Chapter 4, 4.1, P199
![Page 19: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/19.jpg)
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
![Page 20: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/20.jpg)
Chapter 4, 4.2, P205
FIGURE 1 A partition of [a,b] with sample points *ix
![Page 21: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/21.jpg)
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
![Page 22: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/22.jpg)
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.
![Page 23: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/23.jpg)
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
![Page 24: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/24.jpg)
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 )(
![Page 25: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/25.jpg)
Chapter 4, 4.2, P206
drrfdttfdxxfb
a
b
a
ba )()()(
![Page 26: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/26.jpg)
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
![Page 27: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/27.jpg)
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
![Page 28: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/28.jpg)
Chapter 4, 4.2, P208
![Page 29: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/29.jpg)
Chapter 4, 4.2, P208
![Page 30: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/30.jpg)
Chapter 4, 4.2, P208
![Page 31: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/31.jpg)
Chapter 4, 4.2, P208
![Page 32: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/32.jpg)
Chapter 4, 4.2, P210
![Page 33: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/33.jpg)
Chapter 4, 4.2, P211
![Page 34: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/34.jpg)
Chapter 4, 4.2, P211
MIDPOINT RULE
n
i
niba xf‧‧‧xfxxxfdxxf
11 )]()([)()(
where
n
abx
and
],1[int)(2
11 iiiii xxofmidpoxxx
![Page 35: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/35.jpg)
Chapter 4, 4.2, P212
dxxfdxxf ba
ab )()(
![Page 36: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/36.jpg)
Chapter 4, 4.2, P212
0)( dxxfaa
![Page 37: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/37.jpg)
Chapter 4, 4.2, P213
![Page 38: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/38.jpg)
Chapter 4, 4.2, P213
![Page 39: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/39.jpg)
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
![Page 40: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/40.jpg)
Chapter 4, 4.2, P214
![Page 41: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/41.jpg)
Chapter 4, 4.2, P214
dxxfdxxfdxxf ba
bc
ca )()()([.5
![Page 42: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/42.jpg)
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
![Page 43: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/43.jpg)
Chapter 4, 4.2, P215
![Page 44: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/44.jpg)
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
![Page 45: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/45.jpg)
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
![Page 46: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/46.jpg)
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.
![Page 47: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/47.jpg)
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 )()(
![Page 48: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/48.jpg)
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 )(
![Page 49: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/49.jpg)
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
![Page 50: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/50.jpg)
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.
![Page 51: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/51.jpg)
Chapter 4, 4.3, P222
NET CHANGE THEOREM The integral of a rate of change is the net change:
)()()(' aFbFdxxFba
![Page 52: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/52.jpg)
Chapter 4, 4.4, P227
The Fundamental Theorem deals with functions defined by an equation of the from
dttfxg xa )()(
![Page 53: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/53.jpg)
Chapter 4, 4.4, P227
![Page 54: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/54.jpg)
Chapter 4, 4.4, P227
![Page 55: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/55.jpg)
Chapter 4, 4.4, P227
![Page 56: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/56.jpg)
Chapter 4, 4.4, P227
![Page 57: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/57.jpg)
Chapter 4, 4.4, P229
![Page 58: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/58.jpg)
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.
![Page 59: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/59.jpg)
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
![Page 60: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/60.jpg)
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
![Page 61: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/61.jpg)
Chapter 4, 4.4, P232
we define the average value of f on the interval [a,b] as
dxxfab
f baave )(
1
![Page 62: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/62.jpg)
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
![Page 63: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/63.jpg)
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
![Page 64: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/64.jpg)
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
![Page 65: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/65.jpg)
Chapter 4, 4.4, P235
![Page 66: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/66.jpg)
Chapter 4, 4.4, P235
![Page 67: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/67.jpg)
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 )()('))((
![Page 68: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/68.jpg)
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 )()('))(( )(
)(
![Page 69: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/69.jpg)
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
![Page 70: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/70.jpg)
Chapter 4, 4.5, P240
![Page 71: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/71.jpg)
Chapter 4, 4.5, P240
![Page 72: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062803/56649f1b5503460f94c30a1e/html5/thumbnails/72.jpg)
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