...
...
9 Integral Calculus
1. Integration . . . 9-1 2. Elementary Operations . . . 9-1 3. Integration by Parts . . . 9-2 4. Separation of Terms . . . 9-3 5. Double and Higher-Order Integrals . . . 9-3 6. Initial Values . . . 9-4 7. Definite Integrals . . . 9-4 8. Average Value . . . 9-4 9. Area . . . 9-4 10. Arc Length . . . 9-5 11. Pappus’ Theorems . . . 9-5 12. Surface of Revolution . . . 9-5 13. Volume of Revolution . . . 9-6 14. Moments of a Function . . . 9-6 15. Fourier Series . . . 9-6 16. Fast Fourier Transforms . . . 9-8 17. Integral Functions . . . 9-8
1. INTEGRATION
Integration is the inverse operation of differentiation.
For that reason, indefinite integrals are sometimes referred to as antiderivatives.1 Although expressions can be functions of several variables, integrals can only be taken with respect to one variable at a time. The differential term (dx in Eq. 9.1) indicates that variable.
In Eq. 9.1, the function f0ðxÞ is the integrand, and x is the variable of integration.
Z
f0ðxÞdx ¼ f ðxÞ þ C 9:1 While most of a function, fðxÞ, can be “recovered”
through integration of its derivative, f0ðxÞ, a constant term will be lost. This is because the derivative of a constant term vanishes (i.e., is zero), leaving nothing to recover from. A constant of integration, C, is added to the integral to recognize the possibility of such a constant term.
2. ELEMENTARY OPERATIONS
Equation 9.2 through Eq. 9.8 summarize the elemen-tary integration operations on polynomials and exponentials.2
Equation 9.2 and Eq. 9.3 are particularly useful. (C and k represent constants. fðxÞ and gðxÞ are functions of x.)
Z
k dx¼ kx þ C 9:2
Z
xmdx¼ xmþ1
mþ 1þ C ½m 6¼ 1 9:3
Z 1
xdx¼ ln jxj þ C 9:4
Z
ekxdx¼ekx
k þ C 9:5
Z
xekxdx¼ekxðkx 1Þ
k2 þ C 9:6
Z
kaxdx¼ kax
a ln kþ C 9:7
Z
ln x dx¼ x ln x x þ C 9:8
Equation 9.9 through Eq. 9.20 summarize the elemen-tary integration operations on transcendental functions.
Z
sin x dx¼ cos x þ C 9:9
Z
cos x dx¼ sin x þ C 9:10
Z
tan x dx¼ ln jsec xj þ C
¼ ln jcos xj þ C 9:11 Z
cot x dx¼ ln jsin xj þ C 9:12
Z
sec x dx¼ lnjsec x þ tan xj þ C
¼ ln
tan x 2þ p4
þ C 9:13
Z
csc x dx¼ ln csc x cot xj j þ C
¼ ln tanx 2
þ C 9:14
Z dx
k2þ x2¼1
karctan x
kþ C 9:15
Z dx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 x2
p ¼ arcsinx
kþ C ½k2> x2 9:16
1The difference between an indefinite and definite integral (covered in Sec. 9.7) is simple: An indefinite integral is a function, while a definite integral is a number.
2More extensive listings, known as tables of integrals, are widely avail-able. (See App. 9.A.)
Backgroundand Support
...
Z dx
x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 k2
p ¼1
karcsecx
kþ C ½x2> k2 9:17
Z
sin2x dx¼12x14sin 2xþ C 9:18
Z
cos2x dx¼12x14sin 2xþ C 9:19 Z
tan2x dx¼ tan x x þ C 9:20
Equation 9.21 through Eq. 9.26 summarize the elemen-tary integration operations on hyperbolic transcenden-tal functions. Integrals of hyperbolic functions are not completely analogous to those of the regular transcen-dental functions.
Z
sinh x dx¼ cosh x þ C 9:21
Z
cosh x dx¼ sinh x þ C 9:22 Z
tanh x dx¼ ln jcosh xj þ C 9:23
Z
coth x dx¼ ln jsinh xj þ C 9:24
Z
sech x dx¼ arctanðsinh xÞ þ C 9:25
Z
csch x dx¼ ln tanhx 2
þ C 9:26 Equation 9.27 through Eq. 9.30 summarize the elemen-tary integration operations on functions and combina-tions of funccombina-tions.
Z
kfðxÞdx ¼ kZ
fðxÞdx 9:27
Z fðxÞ þ gðxÞ dx¼Z
fðxÞdx þZ
gðxÞdx 9:28
Z f0ðxÞ
fðxÞdx¼ ln jf ðxÞj þ C 9:29 Z
fðxÞdgðxÞ ¼ f ðxÞZ
dgðxÞ Z
gðxÞdf ðxÞ þ C
¼ f ðxÞgðxÞ Z
gðxÞdf ðxÞ þ C 9:30
Example 9.1
Find the integral with respect to x of 3x2þ13x 7 ¼ 0
Solution
This is a polynomial function, and Eq. 9.3 can be applied to each of the three terms.
Z
3x2þ13x 7
dx¼ x3þ16x2 7x þ C
3. INTEGRATION BY PARTS
Equation 9.30, repeated here, is known as integration by parts. fðxÞ and gðxÞ are functions. The use of this method is illustrated by Ex. 9.2.
Z
fðxÞdgðxÞ ¼ f ðxÞgðxÞ Z
gðxÞdf ðxÞ þ C 9:31
Example 9.2
Find the following integral.
Z
x2exdx
Solution
x2ex is factored into two parts so that integration by parts can be used.
fðxÞ ¼ x2 dgðxÞ ¼ exdx dfðxÞ ¼ 2x dx
gðxÞ ¼ Z
dgðxÞ ¼ Z
exdx ¼ ex
From Eq. 9.31, disregarding the constant of integration (which cannot be evaluated),
Z
fðxÞdgðxÞ ¼ f ðxÞgðxÞ Z
gðxÞdf ðxÞ Z
x2exdx¼ x2exZ
exð2xÞdx
The second term is also factored into two parts, and integration by parts is used again. This time,
fðxÞ ¼ x dgðxÞ ¼ exdx dfðxÞ ¼ dx
gðxÞ ¼ Z
dgðxÞ ¼ Z
exdx ¼ ex
BackgroundandSupport
... Then, the complete integral is
Z
x2exdx¼ x2ex 2ðxex exÞ þ C
¼ exðx2 2x þ 2Þ þ C
4. SEPARATION OF TERMS
Equation 9.28 shows that the integral of a sum of terms is equal to a sum of integrals. This technique is known as separation of terms. In many cases, terms are easily separated. In other cases, the technique of partial frac-tions can be used to obtain individual terms. These techniques are illustrated by Ex. 9.3 and Ex. 9.4.
Example 9.3
Find the following integral.
Z ð2x2þ 3Þ2
Find the following integral.
Z 3xþ 2 3x 2dx Solution
The integrand is larger than 1, so use long division to simplify it.
5. DOUBLE AND HIGHER-ORDER INTEGRALS
A function can be successively integrated. (This is anal-ogous to successive differentiation.) A function that is integrated twice is known as a double integral; if inte-grated three times, it is a triple integral; and so on.
Double and triple integrals are used to calculate areas and volumes, respectively.
The successive integrations do not need to be with respect to the same variable. Variables not included in the integration are treated as constants.
There are several notations used for a multiple integral, particularly when the product of length differentials represents a differential area or volume. A double inte-gral (i.e., two successive integrations) can be repre-sented by one of the following notations.
ZZ
A triple integral can be represented by one of the follow-ing notations.
Find the following double integral.
ZZ
...
...
...
...
So, ZZ
ðx2þ y3xÞdx dy ¼13yx3þ18y4x2þ C1yþ C2
6. INITIAL VALUES
The constant of integration, C, can be found only if the value of the function fðxÞ is known for some value of x0. The value fðx0Þ is known as an initial value or initial condition. To completely define a function, as many initial values, fðx0Þ, f0ðx0Þ, f00ðx0Þ, and so on, as there are integrations are needed.
Example 9.6
It is known that fðxÞ ¼ 4 when x ¼ 2 (i.e., the initial value is fð2Þ ¼ 4). Find the original function.
Z
ð3x3 7xÞdx
Solution The function is
fðxÞ ¼Z
ð3x3 7xÞdx ¼34x472x2þ C
Substituting the initial value determines C.
4¼ 34
ð2Þ4 72
ð2Þ2þ C 4¼ 12 14 þ C
C¼ 6 The function is
fðxÞ ¼34x472x2þ 6
7. DEFINITE INTEGRALS
A definite integral is restricted to a specific range of the independent variable. (Unrestricted integrals of the types shown in all preceding examples are known as indefinite integrals.) A definite integral restricted to the region bounded by lower and upper limits (also known as bounds), x1and x2, is written as
Z x2 x1
fðxÞdx
Equation 9.32 indicates how definite integrals are evaluated. It is known as the fundamental theorem of calculus.
Z x2 x1
f0ðxÞdx ¼ f ðxÞx2
x1
¼ f ðx2Þ f ðx1Þ
9:32
A common use of a definite integral is the calculation of work performed by a force, F, that moves an object from position x1to x2.
W¼ Z x2
x1
F dx 9:33
Example 9.7
Evaluate the following definite integral.
Z p=3
p=4
sin x dx
Solution From Eq. 9.32,
Z p=3
p=4
sin x dx¼ cos x
j
p=3p=4¼ cos p3 cos p4
¼ 0:5 ð0:707Þ
¼ 0:207
8. AVERAGE VALUE
The average value of a function fðxÞ that is integrable over the interval½a; b is
average value¼ 1 b a
Z b a
fðxÞdx 9:34
9. AREA
Equation 9.35 calculates the area, A, bounded by x¼ a, x¼ b, f1ðxÞ above and f2ðxÞ below. (f2ðxÞ ¼ 0 if the area is bounded by the x-axis.) This is illustrated in Fig. 9.1.
A¼ Z b
a
f1ðxÞ f2ðxÞ
dx 9:35
Figure 9.1 Area Between Two Curves
f1(x)
f2(x)
x b a
y
A
BackgroundandSupport
...
...
...
Example 9.8
Find the area between the x-axis and the parabola y¼ x2 in the interval½0; 4.
Referring to Eq. 9.35,
f1ðxÞ ¼ x2
Equation 9.36 gives the length of a curve defined by fðxÞ whose derivative exists in the interval [a, b].
length¼Z b
11. PAPPUS’ THEOREMS3
The first and second theorems of Pappus are:4
. first theorem: Given a curve, C, that does not inter-sect the y-axis, the area of the surface of revolution generated by revolving C around the y-axis is equal to the product of the length of the curve and the circumference of the circle traced by the centroid of curve C.
A¼ length circumference
¼ length 2p radius 9:37 . second theorem: Given a plane region, R, that does not intersect the y-axis, the volume of revolution generated by revolving R around the y-axis is equal
to the product of the area and the circumference of the circle traced by the centroid of area R.
V¼ area circumference
¼ area 2p radius 9:38
12. SURFACE OF REVOLUTION
The surface area obtained by rotating fðxÞ about the x-axis is
The surface area obtained by rotating fðyÞ about the y-axis is about the x-axis. What is the surface of revolution?
Solution
The surface of revolution is
x
3This section is an introduction to surfaces and volumes of revolution but does not involve integration.
4Some authorities call the first theorem the second and vice versa.
Backgroundand Support
...
...
...
13. VOLUME OF REVOLUTION
The volume obtained by rotating fðxÞ about the x-axis is given by Eq. 9.41. f2ðxÞ is the square of the function, not the second derivative. Equation 9.41 is known as the method of discs.
V ¼ pZ x¼b x¼a
f2ðxÞdx 9:41
The volume obtained by rotating fðxÞ about the y-axis can be found from Eq. 9.41 (i.e., using the method of discs) by rewriting the limits and equation in terms of y, or alternatively, the method of shells can be used, result-ing in the second form of Eq. 9.42.
V¼ pZ y¼d y¼c
f2ðyÞdy
¼ 2pZ x¼b x¼a
xfðxÞdx 9:42
Example 9.10
The curve fðxÞ ¼ x2over the region x¼ ½0; 4 is rotated about the x-axis. What is the volume of revolution?
Solution
The volume of revolution is
x
z y
y x2
x 4
V¼ pZ b a
f2ðxÞdx ¼ pZ 4
0 ðx2Þ2dx ¼ px5 5
j
40¼ p 1024 5 0
¼ 204:8p
14. MOMENTS OF A FUNCTION
The first moment of a function is a concept used in finding centroids and centers of gravity. Equation 9.43 and Eq. 9.44 are for one- and two-dimensional problems,
respectively. It is the exponent of x (1 in this case) that gives the moment its name.
first moment¼Z
xfðxÞdx 9:43
first moment¼ ZZ
xfðx; yÞdx dy 9:44
The second moment of a function is a concept used in finding moments of inertia with respect to an axis.
Equation 9.45 and Eq. 9.46 are for two- and three-dimensional problems, respectively. Second moments with respect to other axes are analogous.
ðsecond momentÞx ¼ZZ
y2fðx; yÞdy dx 9:45
ðsecond momentÞx¼ ZZZ
ðy2þ z2Þf ðx; y; zÞdy dz dx
9:46
15. FOURIER SERIES
Any periodic waveform can be written as the sum of an infinite number of sinusoidal terms, known as harmonic terms (i.e., an infinite series). Such a sum of terms is known as a Fourier series, and the process of finding the terms is Fourier analysis. (Extracting the original wave-form from the series is known as Fourier inversion.) Since most series converge rapidly, it is possible to obtain a good approximation to the original waveform with a limited number of sinusoidal terms.
Fourier’s theorem is Eq. 9.47.5The object of a Fourier analysis is to determine the coefficients anand bn. The constant a0can often be determined by inspection since it is the average value of the waveform.
fðtÞ ¼ a0þa1cos!t þ a2cos 2!t þ
þ b1sin!t þ b2sin 2!t þ 9:47
! is the natural (fundamental) frequency of the wave-form. It depends on the actual waveform period, T.
! ¼2p
T 9:48
To simplify the analysis, the time domain can be nor-malized to the radian scale. The nornor-malized scale is obtained by dividing all frequencies by !. Then the Fourier series becomes
fðtÞ ¼ a0þ a1cos tþ a2cos 2tþ
þ b1sin tþ b2sin 2tþ 9:49
5The independent variable used in this section is t, since Fourier analysis is most frequently used in the time domain.
BackgroundandSupport
The coefficients an and bnare found from the following
While Eq. 9.51 and Eq. 9.52 are always valid, the work of integrating and finding an and bn can be greatly simplified if the waveform is recognized as being sym-metrical. Table 9.1 summarizes the simplifications.
Example 9.11
Find the first four terms of a Fourier series that approx-imates the repetitive step function illustrated.
fðtÞ ¼ 1 0< t < p
This value of 1=2 corresponds to the average value of fðtÞ. It could have been found by observation.
a1¼1
Table 9.1 Fourier Analysis Simplifications for Symmetrical Waveforms
...
16. FAST FOURIER TRANSFORMS
Many mathematical operations are needed to imple-ment a true Fourier transform. While the terms of a Fourier series might be slowly derived by integration, a faster method is needed to analyze real-time data. The fast Fourier transform (FFT) is a computer algorithm implemented in spectrum analyzers (signal analyzers or FFT analyzers) and replaces integration and multiplica-tion operamultiplica-tions with table look-ups and addimultiplica-tions.6 Since the complexity of the transform is reduced, the transformation occurs more quickly, enabling efficient analysis of waveforms with little or no periodicity.7 Using a spectrum analyzer requires choosing the fre-quency band (e.g., 0–20 kHz) to be monitored. (This step automatically selects the sampling period. The lower the frequencies sampled, the longer the sam-pling period.) If they are not fixed by the analyzer, the numbers of time-dependent input variable sam-ples (e.g., 1024) and frequency-dependent output variable values (e.g., 400) are chosen.8There are half as many frequency lines as data points because each line contains two pieces of information—real (ampli-tude) and imaginary (phase). The resolution of the resulting frequency analysis is
resolution¼ frequency bandwidth
no: of output variable values 9:53
17. INTEGRAL FUNCTIONS
Integrals that cannot be evaluated as finite combinations of elementary functions are called integral functions.
These functions are evaluated by series expansion. Some of the more common functions are listed as follows.9 . integral sine function
SiðxÞ ¼Z x
. integral cosine function CiðxÞ ¼Z x
. integral exponential function EiðxÞ ¼Z x
CEin Eq. 9.55 and Eq. 9.56 is Euler’s constant.
CE¼
6Spectrum analysis, also known as frequency analysis, signature analy-sis, and time-series analyanaly-sis, develops a relationship (usually graphi-cal) between some property (e.g., amplitude or phase shift) versus frequency.
7Hours and days of manual computations are compressed into milliseconds.
8Two samples per time-dependent cycle (at the maximum frequency) is the lower theoretical limit for sampling, but the practical minimum rate is approximately 2.5 samples per cycle. This will ensure that alias components (i.e., low-level frequency signals) do not show up in the frequency band of interest.
9Other integral functions include the Fresnel integral, gamma func-tion, and elliptic integral.
BackgroundandSupport
...
...