Numerical Integration - SharpSchool

4.6 Numerical Integration 305

4.6 Numerical Integration

Approximate a definite integral using the Trapezoidal Rule.Approximate a definite integral using Simpson’s Rule.Analyze the approximate errors in the Trapezoidal Rule and Simpson’s Rule.

The Trapezoidal RuleSome elementary functions simply do not have antiderivatives that are elementary functions. For example, there is no elementary function that has any of the followingfunctions as its derivative.

If you need to evaluate a definite integral involving a function whose antiderivative cannot be found, then while the Fundamental Theorem of Calculus is still true, it cannot be easily applied. In this case, it is easier to resort to an approximation technique. Two such techniques are described in this section.

One way to approximate a definite integral is to use trapezoids, as shown inFigure 4.42. In the development of this method, assume that is continuous and positive on the interval So, the definite integral

represents the area of the region bounded by the graph of and the axis, from to First, partition the interval into subintervals, each of width

such that

Then form a trapezoid for each subinterval (see Figure 4.43). The area of the thtrapezoid is

Area of th trapezoid

This implies that the sum of the areas of the trapezoids is

Letting you can take the limit as to obtain

The result is summarized in the next theorem.

� 0 � �b

a

f �x� dx.

� limn→�

� f �a� � f�b��b � a�

2n� lim

n→� �

n

i�1 f�xi� � x

� limn→�

�� f�a� � f �b�� x2

� �n

i�1 f �xi� �x

limn→�

b � a2n � � f�x0� � 2f�x1� � . . . � 2f�xn�1� � f�xn��

n → ��x � �b � a��n,

� b � a2n �� f �x0� � 2 f �x1� � 2 f �x2� � . . . � 2 f �xn�1� � f �xn��.

� b � a2n �� f �x0� � f �x1� � f �x1� � f �x2� � . . . � f �xn�1� � f �xn��

Area � b � an �� f �x0� � f �x1�

2� . . . �

f �xn�1� � f �xn�2

n

� � f �xi�1� � f �xi�2

b � an �.i

i

a � x0 < x1 < x2 < . . . < xn � b.

�x � �b � a��n,n�a, b�x � b.

x � ax-f

�b

a f �x� dx

�a, b�.f

n

3 x 1 � x, x cos x, cos xx

, 1 � x3, sin x2

x

f

x1 x2 x3x0 = a x4 = b

y

The area of the region can be approximated using four trapezoids.Figure 4.42

xx0 x1

b − an

f (x1)

f (x0)

y

The area of the first trapezoid is

Figure 4.43

�f �x0� � f �x1�2 b � a

n �.

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Approximation with the Trapezoidal Rule

Use the Trapezoidal Rule to approximate

Compare the results for and as shown in Figure 4.44.

Solution When and you obtain

When and you obtain

For this particular integral, you could have found an antiderivative and determined thatthe exact area of the region is 2.

� 1.974.

��

16 2 � 2 2 � 4 sin �

8� 4 sin

3�

8 �� 2 sin

5�

8� 2 sin

3�

4� 2 sin

7�

8� sin ��

��

0 sin x dx �

�

16 sin 0 � 2 sin �

8� 2 sin

�

4� 2 sin

3�

8� 2 sin

�

2

n � 8, �x � ��8,

� 1.896.

��1 � 2�

4

��

8�0 � 2 � 2 � 2 � 0�

��

0 sin x dx �

�

8 sin 0 � 2 sin �

4� 2 sin

�

2� 2 sin

3�

4� sin ��

n � 4, �x � ��4,

n � 8,n � 4

��

0 sin x dx.

306 Chapter 4 Integration

THEOREM 4.17 The Trapezoidal Rule

Let be continuous on The Trapezoidal Rule for approximating is

Moreover, as the right-hand side approaches �ba f �x� dx.n →�,

�b

a

f �x� dx �b � a

2n � f �x0� � 2 f �x1� � 2 f �x2� � . . . � 2 f �xn�1� � f �xn��.

�ba f �x� dx

�a, b�.f

REMARK Observe that the coefficients in the Trapezoidal Rule have the followingpattern.

1 2 2 2 . . . 2 2 1

TECHNOLOGY Most graphing utilities and computer algebra systems havebuilt-in programs that can be used to approximate the value of a definite integral. Tryusing such a program to approximate the integral in Example 1. How close is your approximation? When you use such a program, you need to be aware of its limitations. Often, you are given no indication of the degree of accuracy of theapproximation. Other times, you may be given an approximation that is completelywrong. For instance, try using a built-in numerical integration program to evaluate

Your calculator should give an error message. Does yours?

�2

�1 1x dx.

π ππ π2 44

3x

1

Four subintervals

y

y = sin x

y = sin x

ππππ π2848

3x

1

Eight subintervals

y

π8

5 π8

7π4

3

Trapezoidal approximationsFigure 4.44


It is interesting to compare the Trapezoidal Rule with the Midpoint Rule given inSection 4.2. For the Trapezoidal Rule, you average the function values at the endpointsof the subintervals, but for the Midpoint Rule, you take the function values of the subinterval midpoints.

Midpoint Rule

Trapezoidal Rule

There are two important points that should be made concerning the TrapezoidalRule (or the Midpoint Rule). First, the approximation tends to become more accurateas increases. For instance, in Example 1, when the Trapezoidal Rule yields anapproximation of 1.994. Second, although you could have used the FundamentalTheorem to evaluate the integral in Example 1, this theorem cannot be used to evaluatean integral as simple as because has no elementary antiderivative. Yet,the Trapezoidal Rule can be applied to estimate this integral.

Simpson’s RuleOne way to view the trapezoidal approximation of a definite integral is to say that oneach subinterval, you approximate by a first-degree polynomial. In Simpson’s Rule,named after the English mathematician Thomas Simpson (1710–1761), you take thisprocedure one step further and approximate by second-degree polynomials.

Before presenting Simpson’s Rule, consider the next theorem for evaluating integrals of polynomials of degree 2 (or less).

f

f

sin x2��0 sin x2 dx

n � 16,n

�b

a

f �x� dx � �n

i�1 f�xi� � f�xi�1�

2 � �x

�b

a

f �x� dx � �n

i�1 f xi � xi�1

2 � � x


THEOREM 4.18 Integral of

If then

�b

a

p�x� dx � b � a6 ��p�a� � 4pa � b

2 � � p�b�.

p�x� � Ax2 � Bx � C,

p �x� � Ax2 � Bx � C

Proof

By expansion and collection of terms, the expression inside the brackets becomes

and you can write

See LarsonCalculus.com for Bruce Edwards’s video of this proof.

�b

a

p�x� dx � b � a6 ��p�a� � 4pa � b

2 � � p�b�.

p�b�4pa � b2 �p�a�

�Aa2 � Ba � C� � 4�Ab � a2 �

2

� Bb � a2 � � C � �Ab2 � Bb � C�

� b � a6 ��2A�a2 � ab � b2� � 3B�b � a� � 6C�

�A�b3 � a3�

3�

B�b2 � a2�2

� C�b � a�

� �Ax3

3�

Bx2

2� Cx

b

a

�b

a

p�x� dx � �b

a

�Ax2� Bx � C� dx


To develop Simpson’s Rule for approximating a definite integral, you againpartition the interval into subintervals, each of width Thistime, however, is required to be even, and the subintervals are grouped in pairs suchthat

On each (double) subinterval , you can approximate by a polynomial ofdegree less than or equal to 2. (See Exercise 47.) For example, on the subinterval

choose the polynomial of least degree passing through the points and as shown in Figure 4.45. Now, using as an approximation of on

this subinterval, you have, by Theorem 4.18,

Repeating this procedure on the entire interval produces the next theorem.

In Example 1, the Trapezoidal Rule was used to estimate In the nextexample, Simpson’s Rule is applied to the same integral.

Approximation with Simpson’s Rule

See LarsonCalculus.com for an interactive version of this type of example.

Use Simpson’s Rule to approximate

Compare the results for

Solution When you have

When you have ��

0 sin x dx � 2.0003.n � 8,

��

0 sin x dx �

�

12 sin 0 � 4 sin �

4� 2 sin

�

2� 4 sin

3�

4� sin �� 2.005.

n � 4,

n � 4 and n � 8.

��

0 sin x dx.

��0 sin x dx.

�a, b�

�b � a

3n � f �x0� � 4 f �x1� � f �x2��.

�2��b � a��n�

6 � p�x0� � 4p �x1� � p�x2��

�x2 � x0

6 �p�x0� � 4px0 � x2

2 � � p�x2�

�x2

x0

f �x� dx � �x2

x0

p�x� dx

fp�x2, y2�,�x1, y1�,�x0, y0�,�x0, x2�,

pf�xi�2, xi�

�xn�2, xn��x2, x4��x0, x2�

a � x0 < x1 < x2 < x3 < x4 < . . . < xn�2 < xn�1 < xn � b.

n�x � �b � a��n.n�a, b�


THEOREM 4.19 Simpson’s Rule

Let be continuous on and let be an even integer. Simpson’s Rule forapproximating is

Moreover, as the right-hand side approaches �ba f �x� dx.n → �,

� 4 f �xn�1� � f �xn��.

�b

a

f �x� dx �b � a

3n � f �x0� � 4 f �x1� � 2 f �x2� � 4 f �x3� � . . .

�ba f �x� dx

n�a, b�f

REMARK In Section 4.2,Example 8, the Midpoint Rulewith approximates

as 2.052. In

Example 1, the TrapezoidalRule with gives anapproximation of 1.896. InExample 2, Simpson’s Rulewith gives an approximation of 2.005. Theantiderivative would produce the true value of 2.

n � 4

n � 4

��

0 sin x dx

n � 4

x

pf

x0 x1 x2 xn

(x0, y0)

(x2, y2)

(x1, y1)

y

Figure 4.45

�x2

x0

p�x� dx � �x2

x0

f �x� dx

REMARK Observe that thecoefficients in Simpson’s Rulehave the following pattern.

1 4 2 4 2 4 . . . 4 2 4 1


Error AnalysisWhen you use an approximation technique, it is important to know how accurate youcan expect the approximation to be. The next theorem, which is listed without proof,gives the formulas for estimating the errors involved in the use of Simpson’s Rule andthe Trapezoidal Rule. In general, when using an approximation, you can think of theerror as the difference between and the approximation.

Theorem 4.20 states that the errors generated by the Trapezoidal Rule andSimpson’s Rule have upper bounds dependent on the extreme values of and in the interval Furthermore, these errors can be made arbitrarily small by increasing provided that and are continuous and therefore bounded in

The Approximate Error in the Trapezoidal Rule

Determine a value of such that the Trapezoidal Rule will approximate the value of

with an error that is less than or equal to 0.01.

Solution Begin by letting and finding the second derivative of

and

The maximum value of on the interval So, by Theorem4.20, you can write

To obtain an error that is less than 0.01, you must choose such that

So, you can choose (because must be greater than or equal to 2.89) and applythe Trapezoidal Rule, as shown in Figure 4.46, to obtain

So, by adding and subtracting the error from this estimate, you know that

1.144 � �1

0 1 � x2 dx � 1.164.

� 1.154.

�1

0 1 � x2 dx �

16

� 1 � 02 � 2 1 � �13�2

� 2 1 � �23�2

� 1 � 12�

nn � 3

n 10012 �2.89100 � 12n2

1��12n2� � 1�100.nE

�E� ��b � a�3

12n2 � f �0�� 1

12n2 �1� �1

12n2.

�0, 1� is � f �0�� 1.� f �x��f �x� � �1 � x2��3�2f��x� � x�1 � x2��1�2

f.f �x� � 1 � x2

�1

0 1 � x2 dx

n

�a, b�.f �4�f n,�a, b�.

f �4��x�f�x�

�ba f �x� dxE


THEOREM 4.20 Errors in the Trapezoidal Rule and Simpson’s Rule

If has a continuous second derivative on then the error in approximating by the Trapezoidal Rule is

Trapezoidal Rule

Moreover, if has a continuous fourth derivative on then the error in approximating by Simpson’s Rule is

Simpson’s Rulea � x � b.�E� ��b � a�5

180n4 �max � f �4��x�� ,

�ba f �x� dx

E�a, b�,f

a � x � b.�E� ��b � a�3

12n2 �max � f �x��,

�ba f �x� dx

E�a, b�,f

FOR FURTHER INFORMATIONFor proofs of the formulas used forestimating the errors involved inthe use of the Midpoint Rule andSimpson’s Rule, see the article“Elementary Proofs of ErrorEstimates for the Midpoint andSimpson’s Rules” by Edward C.Fazekas, Jr. and Peter R. Mercer inMathematics Magazine. To viewthis article, go to MathArticles.com.

x1

1

2

2

y

y = 1 + x2

n = 3

Figure 4.46

1.144 � �1

0 1 � x2 dx � 1.164

TECHNOLOGY If you haveaccess to a computer algebrasystem, use it to evaluate thedefinite integral in Example 3.You should obtain a value of

(The symbol “ln” represents the natural logarithmic function,which you will study in Section 5.1.)

� 1.14779.

�12

� 2 � ln �1 � 2��

�1

0 1 � x2 dx



Using the Trapezoidal Rule and Simpson’s Rule InExercises 1–10, use the Trapezoidal Rule and Simpson’s Ruleto approximate the value of the definite integral for the givenvalue of Round your answer to four decimal places and compare the results with the exact value of the definite integral.

1. 2.

3. 4.

5. 6.

7. 8.

9.

10.

Using the Trapezoidal Rule and Simpson’s Rule InExercises 11–20, approximate the definite integral using theTrapezoidal Rule and Simpson’s Rule with Comparethese results with the approximation of the integral using agraphing utility.

11. 12.

13. 14.

15. 16.

17. 18.

19.

20.

Estimating Errors In Exercises 23–26, use the error formulas in Theorem 4.20 to estimate the errors in approximatingthe integral, with using (a) the Trapezoidal Rule and (b) Simpson’s Rule.

23. 24.

25. 26.

Estimating Errors In Exercises 27–30, use the error formulas in Theorem 4.20 to find such that the error in theapproximation of the definite integral is less than or equal to0.00001 using (a) the Trapezoidal Rule and (b) Simpson’s Rule.

27. 28.

29. 30.

Estimating Errors Using Technology In Exercises 31–34,use a computer algebra system and the error formulas to find

such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) theTrapezoidal Rule and (b) Simpson’s Rule.

31. 32.

33. 34.

35. Finding the Area of a Region Approximate the area ofthe shaded region using

(a) the Trapezoidal Rule with

(b) Simpson’s Rule with

Figure for 35 Figure for 36

36. Finding the Area of a Region Approximate the area ofthe shaded region using

(a) the Trapezoidal Rule with

(b) Simpson’s Rule with

37. Area Use Simpson’s Rule with to approximate thearea of the region bounded by the graphs of

and x � ��2.x � 0,y � 0,y � x cos x,

n � 14

n � 8.

n � 8.

x2 4 6 8 10

2

4

6

8

10

y

x1 2 3 4 5

2

4

6

8

10

y

n � 4.

n � 4.

�1

0 sin x2 dx�1

0 tan x2 dx

�2

0 �x � 1�2�3 dx�2

0 1 � x dx

n

��2

0 sin x dx�2

0 x � 2 dx

�1

0

11 � x

dx�3

1 1x dx

n

��

0 cos x dx�4

2

1�x � 1�2 dx

�5

3 �5x � 2� dx�3

1 2x3 dx

n � 4,

f �x� � �sin x ,

x

1,

x > 0

x � 0��

0 f �x� dx,

��4

0 x tan x dx

��2

0 1 � sin 2 x dx�3.1

3 cos x2 dx

� ��4

0 tan x2 dx� ��2

0 sin x2 dx

��

��2 x sin x dx�1

0 x 1 � x dx

�2

0

1

1 � x3 dx�2

0 1 � x3 dx

n � 4.

�2

0 x x2 � 1 dx, n � 4

�1

0

2�x � 2�2 dx, n � 4

�4

1 �4 � x2� dx, n � 6�9

4 x dx, n � 8

�8

0 3 x dx, n � 8�3

1 x3 dx, n � 6

�3

2 2x2 dx, n � 4�2

0 x3 dx, n � 4

�2

1 x2

4� 1� dx, n � 4�2

0 x2 dx, n � 4

n.

4.6 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

WRITING ABOUT CONCEPTS21. Polynomial Approximations The Trapezoidal Rule

and Simpson’s Rule yield approximations of a definiteintegral based on polynomial approximations of

What is the degree of the polynomials used for each?

22. Describing an Error Describe the size of the errorwhen the Trapezoidal Rule is used to approximate

when is a linear function. Use a graph toexplain your answer.

f �x��ba f �x� dx

f.�b

a f �x� dx



38. Circumference The elliptic integral

gives the circumference of an ellipse. Use Simpson’s Rulewith to approximate the circumference.

41. Work To determine the size of the motor required to operate a press, a company must know the amount of workdone when the press moves an object linearly 5 feet. The variable force to move the object is

where is given in pounds and gives the position of the unitin feet. Use Simpson’s Rule with to approximate thework (in foot-pounds) done through one cycle when

42. Approximating a Function The table lists several measurements gathered in an experiment to approximate anunknown continuous function

(a) Approximate the integral

using the Trapezoidal Rule and Simpson’s Rule.

(b) Use a graphing utility to find a model of the formfor the data. Integrate the

resulting polynomial over and compare the resultwith the integral from part (a).

Approximation of Pi In Exercises 43 and 44, useSimpson’s Rule with to approximate using the givenequation. (In Section 5.7, you will be able to evaluate the integral using inverse trigonometric functions.)

43. 44.

45. Using Simpson’s Rule Use Simpson’s Rule withand a computer algebra system to approximate in the

integral equation

46. Proof Prove that Simpson’s Rule is exact when approx-imating the integral of a cubic polynomial function, and demonstrate the result with for

47. Proof Prove that you can find a polynomial

that passes through any three points andwhere the ’s are distinct.xi�x3, y3�,

�x2, y2�,�x1, y1�,

p�x� � Ax2 � Bx � C

�1

0 x3 dx.

n � 4

�t

0 sin x dx � 2.

tn � 10

� � �1

0

41 � x2 dx� � �1�2

0

6 1 � x2

dx

�n � 6

�0, 2�y � ax3 � bx2 � cx � d

�2

0 f �x� dx

x 1.25 1.50 1.75 2.00

y 7.25 7.64 8.08 8.14

x 0.00 0.25 0.50 0.75 1.00

y 4.32 4.36 4.58 5.79 6.14

y � f �x�.

W � �5

0 F�x� dx.

Wn � 12

xF

F�x� � 100x 125 � x3

n � 8

8 3 ��2

0 1 �

23 sin2 � d�

Use the Trapezoidal Ruleto estimate the number of square meters of land,where and are meas-ured in meters, as shownin the figure. The land isbounded by a stream andtwo straight roads thatmeet at right angles.

x

150

100

50

200 400 600 800 1000

Road

Road

Stream

y

x 600 700 800 900 1000

y 95 88 75 35 0

x 0 100 200 300 400 500

y 125 125 120 112 90 90

yx

39. Surveying

40. HOW DO YOU SEE IT? The function isconcave upward on the interval and the func-tion is concave downward on the interval

(a) Using the Trapezoidal Rule with , which integral would be overestimated? Which integralwould be underestimated? Explain your reasoning.

(b) Which rule would you use for more accurateapproximations of and theTrapezoidal Rule or Simpson’s Rule? Explain yourreasoning.

�20 g�x� dx,�2

0 f �x� dx

n � 4

1 2

2

1

3

4

5g(x)

x

y

1 2

2

3

4

5 f(x)

x

y

�0, 2�.g�x��0, 2�

f �x�

Henryk Sadura/Shutterstock.com


Section 4.6 (page 310)

Trapezoidal Simpson’s Exact1. 2.7500 2.6667 2.66673. 4.2500 4.0000 4.00005. 20.2222 20.0000 20.00007. 12.6640 12.6667 12.66679. 0.3352 0.3334 0.3333

Trapezoidal Simpson’s Graphing Utility11. 3.2833 3.2396 3.241313. 0.3415 0.3720 0.392715. 0.5495 0.5483 0.549317.19. 0.1940 0.1860 0.185821. Trapezoidal: Linear (1st-degree) polynomials

Simpson’s: Quadratic (2nd-degree) polynomials23. (a) 1.500 (b) 0.000 25. (a) (b)27. (a) (b) 29. (a) (b)31. (a) (b) 33. (a) (b)

35. (a) 24.5 (b) 25.67 37. 0.701 39.41. 10,233.58 ft-lb 43. 3.1416 45. 2.477 47. Proof

89,250 m2

n � 48n � 643n � 12n � 130n � 8n � 77n � 26n � 366

112

14

�0.0977�0.0977�0.0975

Answers to Odd-Numbered Exercises A41

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