Visualizing Techniques of Integration
Visualizing Techniques of IntegrationOne of the guiding
principles of the calculus reform movement over the past two
decades has been the notion of approaching every concept and
technique from graphical and numerical as well as symbolic points
of view. However, the implementation of this principle invariably
hits a brick wall when first year calculus courses reach the point
where techniques of integration rear their ugly heads and the
presentation devolves to exercises in symbolic manipulation
exclusively.
In this article, we look at several of the standard techniques
of integration and attempt to introduce graphical and numerical
perspectives that serve to provide different and in some ways
deeper insights into the methods.
Integration via Trig Substitutions
We begin by considering the use of trig substitutions to
evaluate antiderivatives involving expressions of the form a2 b2x2
or a2 + b2x2, particularly when the term arises inside a square
root or is raised to some other rational or integer power. Thus,
the substitution x = a/b sin transforms the expression a2 b2x2 into
a2 cos2using the Pythagorean identity. For instance, substituting x
= 3/2 sin so that dx = 3/2 cos dlets us evaluate
This is essentially what would be expected of students in
calculus. Typically, most of us would demonstrate perhaps three
such integrals, followed by one or two definite integrals that
introduce the need either to change the limits of integration to
reflect each successive transformation or to ignore the limits of
integration until the final integral evaluation is completed and
then reverse each substitution to return to an expression in the
original variable and apply the original limits of integration.
Thus, suppose we consider
since Sin-1(2/3) = 0.7297, correct to four decimal places.
But, obviously, there is no graphical or numerical
interpretation involved that emphasizes the conceptual side of what
is happening. Well, what is actually happening? We have a function
whose antiderivative is to be found and, to do so, we transform it
into a different function, involving a different variable whose
antiderivative is much easier (in this case, trivial) to find. Lets
look at this situation visually. In Figure 1a, we show the graph of
the original integrand
on the interval [-1, 1] of x-values and in Figure 1b, we show
the graph of the transformed integrand h() = on the corresponding
interval [-0.7297, 0.7297] of values.
Ideally, students should realize that, because the initial and
final definite integrals are equal, the areas under the
corresponding curves are equal (despite the different coordinate
systems and the different limits of integration) and vice versa.
Understanding this conceptually can totally transform a process
that is performed mechanically into one that makes sense. The above
example provides a simple way to look at the area interpretation
easily, at least for the end integral, because the curve is just a
horizontal line with height , so the region is a rectangle whose
base is 2 0.7297 and the resulting area is 0.7297. The key, then,
is to find the area under the original curve without depending on
the trig substitution. Presuming that students have seen some
numerical integration (which is usually introduced fairly early in
more modern calculus classes), we can easily estimate the value of
the original definite integral. For instance, using Riemann sums
with n = 100 subdivisions, we find that the Left-Hand Sum = 0.72975
and the Right-Hand Sum = 0.72978, both of which are very close to
the area of the rectangle. This is usually adequate to convince
students that the two regions have equal areas; if more is needed,
it is simple to use more than n = 100 subdivisions with Riemann
sums or a more accurate numerical integration technique, such as
the trapezoid rule or even Simpsons rule.
Before proceeding, however, there is one important detail that
needs mentioning if we are to use any numerical integration
technique. Suppose that we opt to subdivide the interval [-1, 1]
into four uniformly spaced subintervals, so x = and the points of
the subdivision are x = -1, -0.5, 0, 0.5 and 1. Lets see the
effects on the final integral in terms of . At first thought, one
might think that the interval [-0.7297, 0.7297] of values is
likewise subdivided into four uniformly spaced subintervals with =
(0.7297 (-0.7297))/4 = 0.36485. However, the relationship between x
and is via the trig substitution x = 3/2 sin , so that =
Sin-1(2x/3). Therefore, when x0 = -1, we have 1 = Sin-1(2(-1)/3) (
-0.7297; when x1 = -0.5, we have 1 = Sin-1(2(-0.5)/3) ( -0.3398;
when x2 = 0, we have 2 = Sin-1(2(0)/3) = 0; when x3 = 0.5, we have
3 = Sin-1(2(0.5)/3) ( 0.3398; and, of course, 4 = Sin-1(2(1)/3) (
0.7297. Thus, the subdivision of the -interval is not uniform.
Needless to say, the same will be true for any number n of
subdivisions because the transformation involves a nonlinear
function.
However, a little deeper reflection will convince you that the
areas of each of the corresponding subdivisions must be equal. For
instance, if the original integral were taken over the interval
[-1, -0.5], then the same argument used above tells us that the
corresponding integral with from -0.7297 to 1 = -0.3398 would have
to be the same.
To facilitate investigation of these ideas (as well as many
other ideas in calculus), the author has developed dynamic
investigatory spreadsheets in Excel that can be downloaded from his
website [1]. The one associated with integration via trig
substitutions allows the user, either an instructor performing an
in-class demonstration or a student pursuing independent
explorations, to use sliders to vary the coefficients a and b in
the expression
and immediately see the effects graphically and numerically. The
user also has the choice of the x-interval being displayed. In
particular, the spreadsheet shows both the graph of the original
function over the desired x-interval and the corresponding graph of
the transformed function in terms of over the resulting -interval.
In addition, it allows the user to control a tracing point via a
slider to highlight the area under the curve from the initial point
to the tracing point (as shown in Figure 1a), as well as the graph
of the area function. The spreadsheet also shows the graph of the
transformed function in terms of and highlights the area under it
from the initial -value to the corresponding tracing point, as
shown in Figure 1b. In this way, students can get a visceral image
of precisely what is happening as a result of the substitution. The
spreadsheet also displays the area traced out in the original graph
as well as the area traced out in the transformed graph; the former
is found using the trapezoid rule and any discrepancies are very
minor and are the results of the errors in the numerical
integration.
In a comparable way, we use the trig substitution x = a/b tan
when an integral involves an expression of the form a2 + b2x2. The
substitution transforms the expression into
For instance, to evaluate , we let x = 3/2 tan so that dx = 3/2
sec2dand therefore
The final expression can then be evaluated as ln sec + tan +
C.
As with the Excel spreadsheet for the sine substitution, the
comparable module for the tangent substitution uses sliders to vary
the coefficients a and b in the expression
,
so that students can immediately see the effects both
graphically and numerically. The user also has the choice of the
x-interval to be displayed and a tracing point. In particular, the
spreadsheet shows (1) the graph of the original function across the
desired x-interval (see Figure 2a), (2) the graph of the area
function (see Figure 2b), and (3) the corresponding graph of the
transformed function in terms of across the resulting -interval
(see Figure 2c). The spreadsheet also displays the area traced out
in the original graph as well as the area traced out in the
transformed graph; both are found using the trapezoid rule, though
the subdivisions of the latter area are definitely not uniformly
spaced. Any discrepancies between the two area estimates are very
minor and are the result of the errors in the numerical
integration. Integration by Parts Integration by parts is another
technique that is typically performed in an essentially mechanical
way; the only understanding that is called for involves identifying
which part of the integrand is to be associated with u and which
with dv. Again, though, it is desirable for students to visualize
how the method works and this can also be achieved through the use
of an appropriate dynamic spreadsheet [2]. The one developed by the
author allows for the use of three families of expressions for the
integrand, f(x) = xp ecx, f(x) = xp sin (cx), and f(x) = xp cos
(cx), where the user can select the values of the two parameters
via sliders. For any desired interval [a, b], the spreadsheet
displays (1) the graph of the function with the area highlighted,
(2) the graph of the area function with the tracing point
highlighted, (3) the graph of the term uv with the area under the
curve highlighted, (4) the graph of the term, and (5) the graph of
the difference of the last two expressions, , which is identical
(other than the shading) to the graph for item (1). See Figures
3a-e for the graphs associated with the function f(x) = x2 cos (-x)
on the interval [0, 3].
Integration via the z-Substitution Another technique of
integration is the so-called z-substitution that is used to
integrate rational functions of the sine and cosine. The authors
spreadsheet [3] considers integrals of the form
where the z-substitution leads to
EMBED Equation.DSMT4 and
The resulting transformed integral is then
which involves only a quadratic expression and so can be
integrated in closed form fairly readily after transforming it into
either a sum or difference of squares. With the spreadsheet, the
user can select the three parameters a, b, and c using sliders, as
well as any desired interval and a tracing point. The spreadsheet
then displays (1) the graph of the original function with the area
highlighted, (2) the graph of the area function with the tracing
point highlighted, and (3) the graph of the transformed function
with the area highlighted. For instance, if a = 2, b = 3, and c = 1
and if the interval is [0, 3], then the original integral is
and the transformed integral is
The graph of the original function is shown in Figure 4a and
that of the transformed function in terms of z is in Figure 4b. The
areas highlighted in each have the same area, for each value of the
tracing point x0. However, notice the major differences in the
behavior of the two functions. In particular, the area under the
original curve from x = 0 to x0 = 2.45, as shown, is 0.565 and
encompasses most of the interval from 0 to 3. In comparison, the
area under the transformed curve from z = 0 to z0 = 2.754 is also
0.565, but spans only a small portion of the overall interval of
z-values, which range from z = 0 to z = 14.101. This is due to the
extreme nonlinearity of the transformation.References1. Author,
Integration via Trig Substitution, Excel spreadsheet, URL to be
supplied.
2. Author, Integration by Parts, Excel spreadsheet, URL to be
supplied.3. Author, Integration via z-Substitutions, Excel
spreadsheet, URL to be supplied.
Figures 1a-b: Graphs associated with
Figures 2a-c: Graphs associated with
Figures 3a-e: Graphs associated with
Figures 4a-c: Graphs associated with
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Area "Under" the Original Curve
x
Sine
Visualizing Trig Substitutions
This DIGMath spreadsheet lets you
visualize the process of
Integration with Trig Substitutions
for a function of the form:
What is the value of the constant a?3
What is the value of the constant b?2
Pick a tracing point x0 =0.5
Your subsitution is then x =1.50150
so that dx =1.50
The corresponding transformed point is0.3327745999= q
What is the interval for x for display?It must be within -1.50
to 1.50
xMin =-10The area from x =-1to x =0.5=0.535
xMax =10
This corresponds to:The area under the transformed graph
q Min =-0.730radiansfrom q =-0.730to q =0.333
q Max =0.730radiansis about0.524
Click each item below for suggestions and investigations
Item 1
Item 2
Item 3
Item 4
Item 5
The area =0.340
Sine
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0.34068831340.344573654
00.3452054256
0.34020690870.3458572319
00.3465293881
0.3397432240.3472222222
00.3479360758
0.33929705320.3486713046
00.3494282789
0.33886819970.3502073842
00.3510090217
0.33845647570.3518336091
00.3526815812
0.33806170190.3535533906
00.3544495084
0.33768370770.3553704251
00.3563166514
0.33732233060.357288719
00.358287182
0.33697741630.359312617
00.3603656251
0.3366488180.3614468325
00.3625568919
0.3363363970.3636964837
00.3648663174
0.33604002180.3660671329
00.3672997023
0.33575956830.3685648312
00.3698633607
0.33549491980.3711961693
00.3725641746
0.33524596630.3739683355
00.3754096547
0.33501260510.3768891807
00.3784080109
0.334794740.3799672938
00.3815682324
0.33459228170.3832120872
00.3849001795
0.33440514750.3866338952
00.3884146891
0.3342332610.3902440882
00.3921236972
0.33407655240.3940552031
00.3960403804
0.33393495820.3980810971
00.4001793205
0.33380842120.4023371247
00.404556697
0.33369689020.4068403468
00.4091905135
0.33360032040.4116097761
00.4141008637
0.3335186730.4166666667
00.4193102485
0.33345191510.4220348595
00.4248439516
0.333400020.4277411943
00.4307304923
0.33336296690.4338160052
00.4370021694
0.3333407410.440293721
00.4436957234
0.33333333330.4472135955
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0.333340741
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0.334233261
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0.33479474
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0.3350126051
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0.3352459663
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0.3354949198
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0.3357595683
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0.3360400218
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0.336336397
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0.336648818
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0.3369774163
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0.3373223306
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0.3376837077
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0.3380617019
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0.3384564757
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0.3388681997
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0.3392970532
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0.339743224
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0.3406883134
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0.3411876532
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0.3417051534
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0.3433690181
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0.3439616149
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0.344573654
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0.3452054256
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0.3465293881
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0.3472222222
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0.3479360758
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0.3486713046
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0.3494282789
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0.3502073842
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0.3510090217
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
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0.3518336091
0
0.3518336091
0
0.3518336091
Area "Under" the Original Curve
x
Tangent
000.53478600070
0.00445454660.53478600070
0.0088744938
0.0132609733
0.0176150641
0.0219377966
0.0262301551
0.0304930808
0.0347274748
0.0389342004
0.043114085
0.0472679226
0.0513964758
0.0555004773
0.0595806316
0.0636376168
0.0676720859
0.0716846681
0.0756759702
0.0796465776
0.0835970555
0.08752795
0.0914397889
0.0953330828
0.0992083257
0.1030659961
0.1069065575
0.110730459
0.1145381367
0.1183300132
0.1221064992
0.1258679933
0.1296148833
0.1333475458
0.1370663476
0.1407716452
0.1444637862
0.1481431088
0.151809943
0.1554646103
0.1591074243
0.1627386911
0.1663587098
0.1699677721
0.1735661633
0.1771541623
0.1807320418
0.1843000686
0.187858504
0.1914076037
0.1949476182
0.198478793
0.202001369
0.2055155821
0.2090216642
0.2125198425
0.2160103404
0.2194933773
0.2229691688
0.2264379268
0.2298998599
0.2333551732
0.2368040686
0.240246745
0.2436833981
0.2471142211
0.2505394043
0.2539591353
0.2573735994
0.2607829792
0.2641874553
0.267587206
0.2709824074
0.2743732336
0.277759857
0.2811424479
0.2845211749
0.2878962051
0.2912677039
0.294635835
0.2980007611
0.3013626432
0.3047216411
0.3080779136
0.311431618
0.3147829109
0.3181319476
0.3214788827
0.3248238699
0.3281670619
0.331508611
0.3348486685
0.3381873854
0.341524912
0.344861398
0.348196993
0.3515318459
0.3548661056
0.3581999205
0.3615334391
0.3648668095
0.3682001798
0.3715336984
0.3748675133
0.378201773
0.3815366259
0.3848722209
0.3882087069
0.3915462335
0.3948849504
0.3982250079
0.401566557
0.4049097491
0.4082547362
0.4116016713
0.414950708
0.4183020009
0.4216557053
0.4250119778
0.4283709757
0.4317328578
0.4350977839
0.4384659151
0.4418374138
0.445212444
0.448591171
0.4519737619
0.4553603853
0.4587512116
0.4621464129
0.4655461636
0.4689506397
0.4723600195
0.4757744836
0.4791942146
0.4826193978
0.4860502208
0.4894868739
0.4929295503
0.4963784457
0.499833759
0.5032956921
0.5067644501
0.5102402416
0.5137232785
0.5172137764
0.5207119548
0.5242180368
0.5277322499
0.5312548259
0.5347860007
0.5383260152
0.5418751149
0.5454335503
0.5490015771
0.5525794566
0.5561674556
0.5597658468
0.5633749091
0.5669949278
0.5706261946
0.5742690086
0.5779236759
0.5815905101
0.5852698327
0.5889619737
0.5926672713
0.5963860731
0.6001187356
0.6038656256
0.6076271197
0.6114036057
0.6151954822
0.6190031599
0.6228270615
0.6266676228
0.6305252932
0.6344005361
0.63829383
0.6422056689
0.6461365634
0.6500870413
0.6540576487
0.6580489508
0.662061533
0.6660960021
0.6701529873
0.6742331416
0.6783371431
0.6824656963
0.6866195339
0.6907994185
0.6950061441
0.6992405381
0.7035034638
0.7077958223
0.7121185548
0.7164726456
0.7208591251
0.7252790723
0.7297336189
The Area Function = The Definite Integral
x
Sheet2
0.500.500.500
0.50.50.50.500.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.50.50.5
0.500.5
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
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0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
Area Under the Transformed Graph
q
Visualizing Trig Substitutions
This DIGMath spreadsheet lets you
visualize the process of
Integration with Trig Substitutions
for a function of the form:
What is the value of the constant a?3
What is the value of the constant b?2
Pick a tracing point x0 =2.5
Your substitution is then x =1.50tan q150
so that dx =1.50sec2q dq
The corresponding transformed point is1.0303768265= q
What is the interval for x for display?It must be within -1.95
to 1.95
xMin =-2Must be larger than -1.95The area from x =-2to x
=2.5is1.191
xMax =4
This corresponds to:The area under the transformed graph
q Min =-0.927radiansfrom q =-0.927to q =1.030
q Max =1.212radiansis about1.191
Click each item below for suggestions and investigations
Item 1
Item 2
Item 3
Item 4
Item 5
The area =0.340
1. One of the most common techniques of Integration is based on
the idea of making a trigonometric substitution to simplify the
integrand. This method is particularly helpful when the integrand
contains an expression of the form a2 - b2x2 or a2 + b2x2,
especially if the expression occurs inside a square root or is
raised to some other fractional power. In particular, the
substitution x = a/b sin q is used when the expression a2 - b2x2
occurs and the substitution x = a/b tan q is used with a2 +
b2x2.
3. The two charts at the upper right show the graph of the
function and the graph of the area function as it is swept out
using the slider below the charts. The corresponding interval of q
values is also displayed. Notice that the pattern in the chart to
the right, for this case, is always linear.
4. The chart at the right shows the graph of the corresponding
function of q after the sine tranformation is performed. Also, the
area that is swept out as both functions are traced is also
highlighted. Although the two curves are very different in
appearance, the areas swept out should be the same because the
integrals before and after the substitution are equal. The value
for the area under the transformed graph is shown to the right of
the chart. This is precisely the graphical representation of the
Trig Substitution technique and the area swept out should match
fairly closely (any minor discrepencies are the result of using
slightly different step sizes in the approximations to the area
used, which are based on the Trapezoid Rule.
5. Click on the tab below labeled "Tangent" to see a comparable
investigation when the tangent substitution is used.
2. The trig substitution technique is usally done in a totally
manipulative manner. This DIGMath spreadsheet lets you investigate
the process graphically for the two cases mentioned. The current
sheet involves integrals containing the square root of a2 - b2x2 .
The trig substitution x = a/b sin q is used to simplify the
expression using the fundamental identity sin2q + cos2q = 1 and the
fact that dx = a/b cos q dq. The spreadsheet gives you a choice of
a range of values for the parameters a and b and any desired
interval within [-a/b, a/b] to assure that a2 - b2 x2 is not
negative.
00.200.171498585100
0.20.20193327470.171498585100.3333259262
00.2038931971
0.20193327470.2058798952
00.2078934688
0.20389319710.2099339861
00.2120014816
0.20587989520.2140959521
00.2162173534
0.20789346880.2183655969
00.2205405457
0.20993398610.2227420101
00.2249697436
0.21200148160.2272234383
00.2295027198
0.21409595210.2318071425
00.234136184
0.21621735340.2364892396
00.2388656168
0.21836559690.241264529
00.2436850894
0.22054054570.2461263049
00.2485870691
0.22274201010.2510661566
00.2535622151
0.22496974360.2560737599
00.2585991662
0.22722343830.2611366635
00.263684328
0.22950271980.2662400777
00.2688016653
0.23180714250.2713666737
00.2739325105
0.2341361840.2764964041
00.2790554
0.23648923960.2816063585
00.284145953
0.23886561680.2866706699
00.2891768096
0.2412645290.2916604892
00.2941176471
0.24368508940.2965440486
00.2989352949
0.24612630490.3012868326
00.3035939666
0.24858706910.3058518751
00.3080556266
0.25106615660.3102001996
00.3122805049
0.25356221510.3142914101
00.316227766
0.25607375990.3180844362
00.3198563272
0.25859916620.3215384218
00.3231258125
0.26113666350.3246137366
00.3259976121
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00.3284360068
0.26624007770.3294825849
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00.3333037077
0.27649640410.3333259262
00.333214878
0.27905540.3329709621
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0.28160635850.3320884897
00.3314530669
0.2841459530.330691013
00.3298049714
0.28667066990.3287979746
00.3276734163
0.28917680960.3264350206
00.3250868098
0.29166048920.3236330693
00.3220783132
0.29411764710.3204272477
00.3186847361
0.29654404860.3168557628
00.3149453995
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0.30128683260.3087773098
00.3065927227
0.30359396660.3043523137
00.3020610467
0.30585187510.299723782
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0.30805562660.2949300789
00.2924826943
0.31020019960.2900073953
00.2875083019
0.31228050490.2849893573
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0.31429141010.279906773
00.2773500981
0.3162277660.2747874997
00.2722219937
0.31808443620.2696564124
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0.31985632720.2645354538
00.2619848548
0.32153842180.259443747
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00.2518963609
0.32461373660.2494114598
00.2469444455
0.32599761210.2444965851
00.2420690239
0.32727307350.239662792
00.2372788109
0.32843600680.2349179
00.2325807824
0.32948258490.2302680914
00.2279803763
0.33040930020.2257181077
00.223481683
0.33121299680.2212714319
00.2190876208
0.33189089860.2169304578
00.2148000969
0.3324406360.2126966425
00.2106201531
0.33286026820.208570645
00.2065480962
0.33314830230.2045524492
00.2025836144
0.33330370770.200641473
00.1987258796
0.33332592620.1968366648
00.1949736374
0.3332148780.193136587
00.1913252855
0.33297096210.1895394895
00.1877789415
0.33259505260.1860433724
00.1843325019
0.33208848970.1826460407
00.1809836918
0.33145306690.179345151
00.1777301086
0.3306910130.1761382504
00.1745692581
0.32980497140.1730228107
00.1714985851
0.32879797460.1699962567
00.1685155001
0.32767341630.1670559895
00.1656173999
0.32643502060.1641994066
00.1628016865
0.32508680980.1614239179
00.1600657813
0.32363306930.1587269594
00.1574071375
0.32207831320.1561060037
00.1548232494
0.32042724770.153558569
00.1523116605
0.31868473610.1510822255
00.1498699693
0.31685576280.1486746011
00.147495834
0.31494539950.1463333853
00.145186976
0.3129587720.1440563317
00.142941182
0.31090102830.1418412605
00.1407563055
0.30877730980.1396860592
00.138630268
0.30659272270.137588683
00.136561059
0.30435231370.1355471554
00.1345467355
0.30206104670.1335595671
00.1325854218
0.2997237820.1316240755
00.1306753082
0.29734525870.1297389037
00.1288146499
0.29493007890.1279023387
00.1270017657
0.29248269430.1261127305
00.1252350362
0.29000739530.12436849
00.1235129023
0.28750830190.1226680875
00.1218338633
0.28498935730.1210100509
00.1201964751
0.28245432240.1193929638
00.1185993485
0.2799067730.1178154637
00.1170411472
0.2773500981
0
0.2747874997
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0.2722219937
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0.2696564124
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0.2670934074
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0.2645354538
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0.2619848548
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0.259443747
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0.2569141062
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0.2543977533
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0.2518963609
0
0.2494114598
0
0.2469444455
0
0.2444965851
0
0.2420690239
0
0.239662792
0
0.2372788109
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0.2349179
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0.2325807824
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0.2302680914
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0.2279803763
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0.2257181077
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0.223481683
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0.2212714319
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0.2190876208
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0.2169304578
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0.2148000969
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0.2126966425
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0.2106201531
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0.200641473
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0.1987258796
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0.1968366648
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0.193136587
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0.1913252855
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0.1895394895
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0.1877789415
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0.1860433724
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0.1843325019
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0.1826460407
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0.1809836918
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0.179345151
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0.1777301086
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0.1761382504
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0.1745692581
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0.1714985851
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0.1714985851
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0.1714985851
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0.1714985851
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0.1714985851
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0.1714985851
Area "Under" the Original Curve
x
Area "Under" the Original Curve Curve
001.19119539270
0.00602899911.19119539270
0.0121163962
0.0182629926
0.024469593
0.0307370049
0.0370660369
0.0434574984
0.049912198
0.0564309422
0.0630145344
0.0696637727
0.076379449
0.0831623467
0.0900132391
0.096932887
0.1039220369
0.1109814183
0.1181117411
0.1253136933
0.1325879376
0.1399351085
0.1473558091
0.1548506075
0.1624200331
0.1700645727
0.1777846666
0.185580704
0.1934530189
0.201401885
0.2094275111
0.2175300362
0.225709524
0.2339659577
0.2422992348
0.2507091612
0.2591954458
0.2677576952
0.2763954074
0.2851079668
0.2938946389
0.3027545643
0.3116867545
0.3206900864
0.3297632984
0.338904986
0.3481135985
0.3573874359
0.3667246465
0.3761232252
0.3855810129
0.3950956959
0.4046648073
0.4142857286
0.4239556921
0.4336717853
0.4434309556
0.4532300158
0.463065652
0.4729344309
0.4828328092
0.4927571436
0.5027037021
0.5126686751
0.5226481887
0.5326383172
0.5426350974
0.5526345419
0.5626326539
0.5726254415
0.5826089318
0.5925791849
0.6025323082
0.6124644694
0.6223719092
0.6322509534
0.6420980243
0.6519096508
0.6616824783
0.6714132765
0.6810989472
0.6907365306
0.7003232104
0.7098563178
0.7193333353
0.7287518979
0.7381097949
0.7474049699
0.7566355204
0.765799696
0.7748958964
0.7839226688
0.7928787044
0.8017628345
0.8105740261
0.8193113774
0.8279741129
0.8365615778
0.845073233
0.8535086494
0.8618675024
0.8701495664
0.8783547088
0.8864828849
0.8945341322
0.9025085651
0.9104063697
0.9182277988
0.9259731666
0.9336428445
0.9412372562
0.9487568735
0.9562022121
0.9635738275
0.9708723117
0.9780982889
0.9852524129
0.9923353636
0.9993478438
1.006290577
1.013164304
1.0199697812
1.0267077781
1.0333790748
1.0399844606
1.0465247318
1.0530006901
1.0594131412
1.0657628931
1.0720507551
1.0782775362
1.0844440444
1.0905510854
1.0965994617
1.102589972
1.1085234101
1.1144005647
1.120222218
1.1259891461
1.1317021177
1.1373618942
1.1429692289
1.148524867
1.1540295452
1.1594839912
1.1648889238
1.1702450527
1.1755530781
1.1808136907
1.1860275717
1.1911953927
1.1963178153
1.2013954916
1.206429064
1.2114191648
1.2163664169
1.2212714333
1.2261348174
1.2309571629
1.235739054
1.2404810655
1.2451837626
1.2498477014
1.2544734286
1.2590614821
1.2636123904
1.2681266733
1.2726048418
1.2770473984
1.2814548367
1.2858276421
1.2901662917
1.2944712544
1.298742991
1.3029819545
1.30718859
1.3113633349
1.3155066192
1.3196188653
1.3237004885
1.3277518969
1.3317734914
1.3357656663
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1.3436632995
1.3475695127
1.351447816
1.3552985708
1.3591221324
1.3629188498
1.3666890663
1.3704331192
1.3741513401
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1.3815115842
1.3851542429
1.3887723408
1.3923661824
1.3959360671
1.3994822892
1.4030051384
The Area Function = The Integral
x
The Area Function = The Definite Integral
0.833333333300.833333333300.971825315800
0.82535514240.83333333330.82535514240.971825315801.4240006242
0.817421419900.8174214199
0.80953347330.82535514240.8095334733
0.801692653800.8016926538
0.79390035760.81742141990.7939003576
0.786158027700.7861580277
0.77846715480.80953347330.7784671548
0.770829279300.7708292793
0.76324599210.80169265380.7632459921
0.755718936600.7557189366
0.74824980970.79390035760.7482498097
0.740840363700.7408403637
0.7334924070.78615802770.733492407
0.726207806200.7262078062
0.7189884870.77846715480.718988487
0.711836435600.7118364356
0.70475370010.77082927930.7047537001
0.697742391200.6977423912
0.69080468380.76324599210.6908046838
0.683942817600.6839428176
0.67715909830.75571893660.6771590983
0.67045589800.670455898
0.66383565570.74824980970.6638356557
0.657300878200.6573008782
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0.644498082100.6444980821
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0.632068913300.6320689133
0.62600141990.72620780620.6260014199
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0.61417514690.7189884870.6141751469
0.608422367900.6084223679
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0.597252970800.5972529708
0.59184269680.70475370010.5918426968
0.586553019300.5865530193
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0.576348659900.5763486599
0.57144067450.69080468380.5714406745
0.566666666700.5666666667
0.56203005060.68394281760.5620300506
0.557534254100.5575342541
0.55318271040.67715909830.5531827104
0.5489788500.54897885
0.54492609080.6704558980.5449260908
0.541027828400.5410278284
0.53728742570.66383565570.5372874257
0.533708201600.5337082016
0.53029341980.65730087820.5302934198
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0.52396988890.65085413970.5239698889
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0.50027770070.60842236790.5002777007
0.500044442500.5000444425
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0.501109879300.5011098793
0.50187426490.59184269680.5018742649
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0.50689687750.58138722990.5068968775
0.508636521600.5086365216
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0.512683571500.5126835715
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0.52013887030.56666666670.5201388703
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0.716596895800.7165968958
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0.731057073300.7310570733
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0.8711167800.87111678
0.87932423550.50054414830.8793242355
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0.89584844950.50110987930.8958484495
0.904163210400.9041632104
0.9125117960.50187426490.912511796
0.920893286100.9208932861
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0.971825315800.9718253158
0.98041374490.50689687750.9804137449
0.989028704200.9890287042
0.99766950660.50863652160.9976695066
1.006335486401.0063354864
1.01502599860.51056613461.0150259986
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1.05002116380.51498651551.0500211638
1.05882534501.058825345
1.06765058790.51747248991.0676505879
1.076496374601.0764963746
1.08536220270.52013887031.0853622027
1.094247585201.0942475852
1.10315204960.52298289751.1031520496
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1.13895175980.52919225661.1389517598
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1.15695481520.53255151031.1569548152
1.165980750701.1659807507
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1.184079576901.1840795769
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0.9718253158
0
0.9718253158
0
0.9718253158
0
0.9718253158
0
0.9718253158
0
0.9718253158
0
Area Under the Transformed Graph
q
Area Under the Transformed Graph
SINETANGENT
xf(x)Areaqf(q )Areaxxf(xx)sweep xsweep yqf(q )xxxf(x)Areaqf(q
)Areaxxxxf(xx)sweep xsweep yqf(q )
-10.44721359550-0.72972765620.50-10-10-0.72972765620-20.20-0.9272952180.83333333330-20-20-0.9272952180
-0.990.44369572340.0044545466-0.72081876090.50.0044544477-10.4472135955-10.4472135955-0.72972765620.5-1.970.20193327470.0060289991-0.92002555610.82535514240.0060290522-20.2-20.2-0.9272952180.8333333333
-0.980.4402937210.0088744938-0.71197905490.50.0088743007-0.990-0.990-0.72081876090-1.940.20389319710.0121163962-0.91261438050.81742141990.012116505-1.970-1.970-0.92002555610
-0.970.43700216940.0132609733-0.70320627580.50.0132606902-0.990.4436957234-0.990.4436957234-0.72081876090.5-1.910.20587989520.0182629926-0.90505835680.80953347330.0182631599-1.970.2019332747-1.970.2019332747-0.92002555610.8253551424
-0.960.43381600520.0176150641-0.69449826560.50.0176146953-0.980-0.980-0.71197905490-1.880.20789346880.024469593-0.89735408510.80169265380.0244698218-1.940-1.940-0.91261438050
-1-0.950.43073049230.0219377966-0.68585296460.50.0219373458-0.980.440293721-0.980.440293721-0.71197905490.5-1-1.850.20993398610.0307370049-0.88949810150.79390035760.030737298-1.940.2038931971-1.940.2038931971-0.91261438050.8174214199
1-0.940.42774119430.0262301551-0.67726840460.50.0262296258-0.970-0.970-0.703206275801-1.820.21200148160.0370660369-0.88148687890.78615802770.0370663978-1.910-1.910-0.90505835680
-0.930.42484395160.0304930808-0.66874270320.50.0304924765-0.970.4370021694-0.970.4370021694-0.70320627580.5-1.790.21409595210.0434574984-0.87331682940.77846715480.0434579304-1.910.2058798952-1.910.2058798952-0.90505835680.8095334733
dx
=-0.920.42203485950.0347274748-0.66027405890.50.0347267987-0.960-0.960-0.69449826560dxx
=-1.760.21621735340.049912198-0.86498430610.77082927930.0499127047-1.880-1.880-0.89735408510
0.01-0.910.41931024850.0389342004-0.65186074570.50.0389334553-0.960.4338160052-0.960.4338160052-0.69449826560.50.03-1.730.21836559690.0564309422-0.85648560620.76324599210.0564315274-1.880.2078934688-1.880.2078934688-0.89735408510.8016926538
-0.90.41666666670.043114085-0.64350110880.50.0431132737-0.950-0.950-0.68585296460-1.70.22054054570.0630145344-0.84781697340.75571893660.063015202-1.850-1.850-0.88949810150
-0.890.41410086370.0472679226-0.63519356050.50.0472670479-0.950.4307304923-0.950.4307304923-0.68585296460.5-1.670.22274201010.0696637727-0.83897460220.74824980970.069664527-1.850.2099339861-1.850.2099339861-0.88949810150.7939003576
First
graph-0.880.41160977610.0513964758-0.62693657620.50.05139554-0.940-0.940-0.67726840460First
graph-1.640.22496974360.076379449-0.82995464150.74084036370.0763802943-1.820-1.820-0.88148687890
0.50-0.870.40919051350.0555004773-0.61872869070.50.0554994828-0.940.4277411943-0.940.4277411943-0.67726840460.52.50-1.610.22722343830.0831623467-0.82075319990.7334924070.0831632878-1.820.2120014816-1.820.2120014816-0.88148687890.7861580277
0.50.3535533906-0.860.40684034680.0595806316-0.6105684950.50.0595795806-0.930-0.930-0.668742703202.50.1714985851-1.580.22950271980.0900132391-0.81136635060.72620780620.0900142808-1.790-1.790-0.87331682940
-0.850.4045566970.0636376168-0.60245463340.50.0636365114-0.930.4248439516-0.930.4248439516-0.66874270320.5-1.550.23180714250.096932887-0.80179013770.7189884870.0969340344-1.790.2140959521-1.790.2140959521-0.87331682940.7784671548
Second
graph-0.840.40233712470.0676720859-0.59438580.50.0676709281-0.920-0.920-0.66027405890Second
graph-1.520.2341361840.1039220369-0.79202058310.71183643560.1039232955-1.760-1.760-0.86498430610
0.50-0.830.40017932050.0716846681-0.58636073660.50.0716834598-0.920.4220348595-0.920.4220348595-0.66027405890.52.50-1.490.23648923960.1109814183-0.78205369430.70475370010.1109827938-1.760.2162173534-1.760.2162173534-0.86498430610.7708292793
0.50.5347860007-0.820.39808109710.0756759702-0.578378230.50.0756747131-0.910-0.910-0.651860745702.51.1911953927-1.460.23886561680.1181117411-0.77188547240.69774239120.1181132395-1.730-1.730-0.85648560620
-0.810.39604038040.0796465776-0.57043710940.50.0796452734-0.910.4193102485-0.910.4193102485-0.65186074570.5-1.430.2412645290.1253136933-0.76151192220.69080468380.1253153208-1.730.2183655969-1.730.2183655969-0.85648560620.7632459921
Third
graph-0.80.39405520310.0835970555-0.56253624450.50.0835957058-0.90-0.90-0.64350110880Third
graph-1.40.24368508940.1325879376-0.75092906240.68394281760.1325897009-1.70-1.70-0.84781697340
0.33277459990-0.790.39212369720.08752795-0.55467454350.50.0875265564-0.90.4166666667-0.90.4166666667-0.64350110880.51.03037682650-1.370.24612630490.1399351085-0.74013293680.67715909830.1399370145-1.70.2205405457-1.70.2205405457-0.84781697340.7557189366
0.33277459990.5-0.780.39024408820.0914397889-0.54685095070.50.0914383528-0.890-0.890-0.635193560501.03037682650.9718253158-1.340.24858706910.1473558091-0.72911962710.6704558980.1473578652-1.670-1.670-0.83897460220
-0.770.38841468910.0953330828-0.53906444520.50.0953316055-0.890.4141008637-0.890.4141008637-0.63519356050.5-1.310.25106615660.1548506075-0.71788526680.66383565570.1548528212-1.670.2227420101-1.670.2227420101-0.83897460220.7482498097
x
vert-0.760.38663389520.0992083257-0.53131403910.50.0992068086-0.880-0.880-0.62693657620x
vert-1.280.25356221510.1624200331-0.70642605570.65730087820.1624224124-1.640-1.640-0.82995464150
00-0.750.38490017950.1030659961-0.52359877560.50.1030644403-0.880.4116097761-0.880.4116097761-0.62693657620.500-1.250.25607375990.1700645727-0.69473827620.65085413970.1700671261-1.640.2249697436-1.640.2249697436-0.82995464150.7408403637
00.4472135955-0.740.38321208720.1069065575-0.51591772780.50.1069049642-0.870-0.870-0.6187286907000.3333259262-1.220.25859916620.1777846666-0.6828183110.64449808210.1777874028-1.610-1.610-0.82075319990
-0.730.38156823240.110730459-0.50826999720.50.1107288295-0.870.4091905135-0.870.4091905135-0.61872869070.5-1.190.26113666350.185580704-0.67066266140.63823541460.1855836323-1.610.2272234383-1.610.2272234383-0.82075319990.733492407
q
vert-0.720.37996729380.1145381367-0.50065471240.50.1145364719-0.860-0.860-0.6105684950q
vert-1.160.2636843280.1934530189-0.65826796780.63206891330.1934561487-1.580-1.580-0.81136635060
00-0.710.37840801090.1183300132-0.49307102780.50.1183283142-0.860.4068403468-0.860.4068403468-0.6105684950.500-1.130.26624007770.201401885-0.6456310310.62600141990.2014052264-1.580.2295027198-1.580.2295027198-0.81136635060.7262078062
00.5-0.70.37688918070.1221064992-0.48551812230.50.122104767-0.850-0.850-0.6024546334001.4240006242-1.10.26880166530.2094275111-0.6327488350.62003584130.2094310745-1.550-1.550-0.80179013770
-0.690.37540965470.1258679933-0.47799519850.50.1258662289-0.850.404556697-0.850.404556697-0.60245463340.5-1.070.27136667370.2175300362-0.61961857120.61417514690.2175338324-1.550.2318071425-1.550.2318071425-0.80179013770.718988487
x-axis-0.680.37396833550.1296148833-0.47050148140.50.1296130874-0.840-0.840-0.59438580x-axis-1.040.27393251050.225709524-0.60623766420.60842236790.2257135643-1.520-1.520-0.79202058310
-10-0.670.37256417460.1333475458-0.46303621740.50.1333457194-0.840.4023371247-0.840.4023371247-0.59438580.5-20-1.010.27649640410.2339659577-0.59260379810.6027805940.2339702538-1.520.234136184-1.520.234136184-0.79202058310.7118364356
10-0.660.37119616930.1370663476-0.45559867340.50.1370644914-0.830-0.830-0.5863607366040-0.980.27905540.2422992348-0.57871494490.59725297080.2423037987-1.490-1.490-0.78205369430
-0.650.36986336070.1407716452-0.4481881360.50.1407697601-0.830.4001793205-0.830.4001793205-0.58636073660.5-0.950.28160635850.2507091612-0.56456939370.59184269680.2507140056-1.490.2364892396-1.490.2364892396-0.78205369430.7047537001
theta-axis-0.640.36856483120.1444637862-0.44080391040.50.1444618729-0.820-0.820-0.578378230theta-axis-0.920.2841459530.2591954458-0.55016578040.58655301930.2592005837-1.460-1.460-0.77188547240
-0.72972765620-0.630.36729970230.1481431088-0.43344532010.50.1481411681-0.820.3980810971-0.820.3980810971-0.578378230.5-0.9272952180-0.890.28667066990.2677576952-0.5355031190.58138722990.2677631399-1.460.2388656168-1.460.2388656168-0.77188547240.6977423912
0.72972765620-0.620.36606713290.151809943-0.42611170530.50.1518079755-0.810-0.810-0.570437109401.21202565650-0.860.28917680960.2763954074-0.52058083350.57634865990.2764011726-1.430-1.430-0.76151192220
-0.610.36486631740.1554646103-0.41880242310.50.1554626166-0.810.3960403804-0.810.3960403804-0.57043710940.5-0.830.29166048920.2851079668-0.50539878930.57144067450.2851140668-1.430.241264529-1.430.241264529-0.76151192220.6908046838
-0.60.36369648370.1591074243-0.41151684610.50.1591054051-0.80-0.80-0.56253624450-0.80.29411764710.2938946389-0.48995732630.56666666670.2939010881-1.40-1.40-0.75092906240
-0.590.36255689190.1627386911-0.40425436210.50.1627366471-0.80.3940552031-0.80.3940552031-0.56253624450.5-0.770.29654404860.3027545643-0.47425729050.56203005060.3027613775-1.40.2436850894-1.40.2436850894-0.75092906240.6839428176
q0
=-0.7297276562-0.580.36144683250.1663587098-0.39701437350.50.1663566414-0.790-0.790-0.55467454350q0
=-0.927295218-0.740.29893529490.3116867545-0.45830006640.55753425410.3116939467-1.370-1.370-0.74013293680
q00
=0.7297276562-0.570.36036562510.1699677721-0.38979629650.50.1699656799-0.790.3921236972-0.790.3921236972-0.55467454350.5q00
=1.2120256565-0.710.30128683260.3206900864-0.44208760790.55318271040.3206976731-1.370.2461263049-1.370.2461263049-0.74013293680.6771590983
dq
=0.0072972766-0.560.3593126170.1735661633-0.38259956050.50.1735640479-0.780-0.780-0.54685095070dq
=0.0106966044-0.680.30359396660.3297632984-0.42562246830.548978850.329771295-1.340-1.340-0.72911962710
-0.550.3582871820.1771541623-0.3754236080.50.1771520241-0.780.3902440882-0.780.3902440882-0.54685095070.5-0.650.30585187510.338904986-0.4089078290.54492609080.3389134083-1.340.2485870691-1.340.2485870691-0.72911962710.670455898
-0.540.3572887190.1807320418-0.36826789340.50.1807298814-0.770-0.770-0.53906444520-0.620.30805562660.3481135985-0.39194752550.54102782840.3481224623-1.310-1.310-0.71788526680
-0.530.35631665140.1843000686-0.36113188330.50.1842978865-0.770.3884146891-0.770.3884146891-0.53906444520.5-0.590.31020019960.3573874359-0.37474607220.53728742570.3573967571-1.310.2510661566-1.310.2510661566-0.71788526680.6638356557
a/b
=1.5-0.520.35537042510.187858504-0.35401505510.50.1878563006-0.760-0.760-0.53131403910a/b
=1.5-0.560.31228050490.3667246465-0.3573086830.53370820160.3667344409-1.280-1.280-0.70642605570
-0.510.35444950840.1914076037-0.34691689750.50.1914053793-0.760.3866338952-0.760.3866338952-0.53131403910.5-0.530.31429141010.3761232252-0.33964128950.53029341980.3761335085-1.280.2535622151-1.280.2535622151-0.70642605570.6573008782
-0.50.35355339060.1949476182-0.33983690950.50.1949453734-0.750-0.750-0.52359877560-0.50.3162277660.3855810129-0.32175055440.52704627670.3855918007-1.250-1.250-0.69473827620
-0.490.35268158120.198478793-0.33277459990.50.1984765282-0.750.3849001795-0.750.3849001795-0.52359877560.5-0.470.31808443620.3950956959-0.30364388150.52396988890.3951070037-1.250.2560737599-1.250.2560737599-0.69473827620.6508541397
-0.480.35183360910.202001369-0.32572948730.50.2019990845-0.740-0.740-0.51591772780-0.440.31985632720.4046648073-0.28532942020.52106728080.4046766501-1.220-1.220-0.6828183110
-0.470.35100902170.2055155821-0.31870109960.50.2055132783-0.740.3832120872-0.740.3832120872-0.51591772780.5-0.410.32153842180.4142857286-0.26681606490.51834137190.414298121-1.220.2585991662-1.220.2585991662-0.6828183110.6444980821
-0.460.35020738420.2090216642-0.31168897340.50.2090193414-0.730-0.730-0.50826999720-0.380.32312581250.4239556921-0.24811344910.51579496360.4239686482-1.190-1.190-0.67066266140
-0.450.34942827890.2125198425-0.3046926540.50.2125175011-0.730.3815682324-0.730.3815682324-0.50826999720.5-0.350.32461373660.4336717853-0.22923193330.51343072670.4336853188-1.190.2611366635-1.190.2611366635-0.67066266140.6382354146
-0.440.34867130460.2160103404-0.29771169470.50.2160079807-0.720-0.720-0.50065471240-0.320.32599761210.4434309556-0.21018258670.51125118850.4434450793-1.160-1.160-0.65826796780
-0.430.34793607580.2194933773-0.29074565690.50.2194909997-0.720.3799672938-0.720.3799672938-0.50065471240.5-0.290.32727307350.4532300158-0.19097716280.50925872050.453244742-1.160.263684328-1.160.263684328-0.65826796780.6320689133
-0.420.34722222220.2229691688-0.28379410920.50.2229667735-0.710-0.710-0.49307102780-0.260.32843600680.463065652-0.17162806840.50745552620.463080992-1.130-1.130-0.6456310310
-0.410.34652938810.2264379268-0.27685662780.50.2264355142-0.710.3784080109-0.710.3784080109-0.49307102780.5-0.230.32948258490.4729344309-0.15214832640.50584362980.472950395-1.130.2662400777-1.130.2662400777-0.6456310310.6260014199
-0.40.34585723190.2298998599-0.26993279580.50.2298974302-0.70-0.70-0.48551812230-0.20.33040930020.4828328092-0.13255153230.5044248650.4828494068-1.10-1.10-0.6327488350
-0.390.34520542560.2333551732-0.26302220290.50.2333527267-0.70.3768891807-0.70.3768891807-0.48551812230.5-0.170.33121299680.4927571436-0.11285180430.50320086560.4927743832-1.10.2688016653-1.10.2688016653-0.6327488350.6200358413
-0.380.3445736540.2368040686-0.25612444520.50.2368016055-0.690-0.690-0.47799519850-0.140.33189089860.5027037021-0.09306372870.50217305560.5027215908-1.070-1.070-0.61961857120
-0.370.34396161490.240246745-0.24923912520.50.2402442655-0.690.3754096547-0.690.3754096547-0.47799519850.5-0.110.3324406360.5126686751-0.07320229910.50134264180.512687219-1.070.2713666737-1.070.2713666737-0.61961857120.6141751469
-0.360.34336901810.2436833981-0.2423658510.50.2436809026-0.680-0.680-0.47050148140-0.080.33286026820.5226481887-0.05328285160.50071060620.5226673926-1.040-1.040-0.60623766420
-0.350.34279558510.2471142211-0.23550423670.50.2471117098-0.680.3739683355-0.680.3739683355-0.47050148140.5-0.050.33314830230.5326383172-0.03332099590.50027770070.5326581846-1.040.2739325105-1.040.2739325105-0.60623766420.6084223679
-0.340.34224104890.2505394043-0.22865390170.50.2505368773-0.670-0.670-0.46303621740-0.020.33330370770.5426350974-0.01333254330.50004444250.5426556305-1.010-1.010-0.59260379810
-0.330.34170515340.2539591353-0.22181447050.50.2539565929-0.670.3725641746-0.670.3725641746-0.46303621740.50.010.33332592620.55263454190.00666656790.5000111110.5526557416-1.010.2764964041-1.010.2764964041-0.59260379810.602780594
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