4.6 Numerical Integration 305

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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 Rule y Some elementary functions simply do not have antiderivatives that are elementary functions. For example, there is no elementary function that has any of the following functions as its derivative. cos x Ί3 xΊ1 x, Ίx cos x, , Ί1 x3, sin x2 x If you need to evaluate a definite integral involving a function whose antiderivative f 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. x x = axx x x = b One way to approximate a definite integral is to use n trapezoids, as shown in 0 1 2 3 4 Figure 4.42. In the development of this method, assume that f is continuous and The area of the region can be positive on the interval ͓a, b͔. So, the definite integral approximated using four trapezoids. b Figure 4.42 ͵ ͑ ͒ f x dx a represents the area of the region bounded by the graph of f and the x- axis, from x a to x b. First, partition the interval ͓a, b͔ into n subintervals, each of width x ͑b a͒͞n, such that . . . a x0 < x1 < x2 < < xn b. y Then form a trapezoid for each subinterval (see Figure 4.43). The area of the i th trapezoid is

f ͑x ͒ f ͑x ͒ b a Area of i th trapezoid ΄ i 1 i ΅΂ ΃. 2 n This implies that the sum of the areas of the n trapezoids is f(x0) b a f ͑x ͒ f ͑x ͒ f ͑x ͒ f ͑x ͒ Area ΂ ΃΄ 0 1 . . . n 1 n ΅ f(x1) n 2 2 x b a . . . x x ΂ ΃͓ f ͑x ͒ f ͑x ͒ f ͑x ͒ f ͑x ͒ f ͑x ͒ f ͑x ͔͒ 0 1 2n 0 1 1 2 n 1 n b − a n b a ΂ ΃͓ f ͑x ͒ 2 f ͑x ͒ 2 f ͑x ͒ . . . 2 f ͑x ͒ f ͑x ͔͒. 2n 0 1 2 n 1 n The area of the first trapezoid is f ͑x ͒ f ͑x ͒ b a Letting x ͑b a͒͞n, you can take the limit as n → to obtain ΄ 0 1 ΅΂ ΃. 2 n b a ͓ ͑ ͒ ͑ ͒ . . . ͑ ͒ ͑ ͔͒ Figure 4.43 lim ΂ ΃ f x0 2f x1 2f xn1 f xn n→ 2n ͓ f͑a͒ f ͑b͔͒ x n lim ΄ f ͑xi͒ x΅ → ͚ n 2 i1 ͓ f ͑a͒ f͑b͔͒͑b a͒ n lim lim f͑x ͒ x → → ͚ i n 2n n i1 b 0 ͵ f ͑x͒ dx. a The result is summarized in the next theorem.

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THEOREM 4.17 The Trapezoidal Rule Let f be continuous on ͓a, b͔. The Trapezoidal Rule for approximating ͐b ͑ ͒ a f x dx is b ͵ ͑ ͒ Ϸ b a ͓ ͑ ͒ ͑ ͒ ͑ ͒ . . . ͑ ͒ ͑ ͔͒ f x dx f x0 2 f x1 2 f x2 2 f xn1 f xn . a 2n → ͐b ͑ ͒ Moreover, as n , the right-hand side approaches a f x dx.

REMARK Observe that the coefficients in the Trapezoidal Rule have the following pattern. 1222. . .221

Approximation with the Trapezoidal Rule

y Use the Trapezoidal Rule to approximate y = sin x ͵ sin x dx. 1 0 Compare the results for n 4 and n 8, as shown in Figure 4.44. ͞ x Solution When n 4, x 4, and you obtain πππ π 3 3 4 24 ͵ sin x dx Ϸ ΂sin 0 2 sin 2 sin 2 sin sin ΃ 8 4 2 4 Four subintervals 0 ͑0 Ί2 2 Ί2 0͒ y 8 ͑1 Ί2͒ y = sin x 4 1 Ϸ 1.896. When n 8, x ͞8, and you obtain x 3 π π 3π π 5π 3π 7π π ͵ sin x dx Ϸ sin 0 2 sin 2 sin 2 sin 2 sin 8 4 8 2 8 4 8 ΂ 0 16 8 4 8 2 5 3 7 Eight subintervals 2 sin 2 sin 2 sin sin ΃ Trapezoidal approximations 8 4 8 Figure 4.44 3 ΂2 2Ί2 4 sin 4 sin ΃ 16 8 8 Ϸ 1.974. For this particular integral, you could have found an antiderivative and determined that the exact area of the region is 2.

TECHNOLOGY Most graphing utilities and computer algebra systems have built-in programs that can be used to approximate the value of a definite integral. Try using 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 the approximation. Other times, you may be given an approximation that is completely wrong. For instance, try using a built-in numerical integration program to evaluate 2 1 ͵ dx. 1 x Your calculator should give an error message. Does yours?

It is interesting to compare the Trapezoidal Rule with the Midpoint Rule given in Section 4.2. For the Trapezoidal Rule, you average the function values at the endpoints of the subintervals, but for the Midpoint Rule, you take the function values of the subinterval midpoints. b n xi xi1 ͵ f ͑x͒ dx Ϸ ͚ f ΂ ΃ x Midpoint Rule a i1 2 b n ͑ ͒ ͑ ͒ f xi f xi1 ͵ ͑ ͒ Ϸ Trapezoidal Rule f x dx ͚ ΂ ΃ x a i1 2 There are two important points that should be made concerning the Trapezoidal Rule (or the Midpoint Rule). First, the approximation tends to become more accurate as n increases. For instance, in Example 1, when n 16, the Trapezoidal Rule yields an approximation of 1.994. Second, although you could have used the Fundamental Theorem to evaluate the integral in Example 1, this theorem cannot be used to evaluate ͐ 2 2 an integral as simple as 0 sin x dx because sin x has no elementary antiderivative. Yet, the Trapezoidal Rule can be applied to estimate this integral.

Simpson’s Rule One way to view the trapezoidal approximation of a definite integral is to say that on each subinterval, you approximate f by a first-degree polynomial. In Simpson’s Rule, named after the English mathematician Thomas Simpson (1710–1761), you take this procedure one step further and approximate f by second-degree polynomials. Before presenting Simpson’s Rule, consider the next theorem for evaluating integrals of polynomials of degree 2 (or less).

Ax 2 ؉ Bx ؉ C ؍ THEOREM 4.18 Integral of p ͧxͨ If p͑x͒ Ax2 Bx C, then b b a a b ͵ p͑x͒ dx ΂ ΃΄p͑a͒ 4p΂ ΃ p͑b͒΅. a 6 2

Proof b b ͵ p͑x͒ dx ͵ ͑Ax2 Bx C͒ dx a a Ax3 Bx2 b ΄ Cx΅ 3 2 a A͑b3 a3͒ B͑b2 a2͒ C͑b a͒ 3 2 b a ΂ ΃͓2A͑a2 ab b2͒ 3B͑b a͒ 6C͔ 6 By expansion and collection of terms, the expression inside the brackets becomes b a 2 b a ͑Aa2 Ba C͒ 4΄A΂ ΃ B΂ ΃ C΅ ͑Ab2 Bb C͒ 2 2

a b p͑a͒ 4p΂ ΃ p͑b͒ 2 and you can write b b a a b ͵ p͑x͒ dx ΂ ΃΄p͑a͒ 4p΂ ΃ p͑b͒΅. a 6 2 See LarsonCalculus.com for Bruce Edwards’s video of this proof.

y To develop Simpson’s Rule for approximating a definite integral, you again partition the interval ͓a, b͔ into n subintervals, each of width x ͑b a͒͞n. This (x , y ) 2 2 time, however,n is required to be even, and the subintervals are grouped in pairs such that p f . . . a x0 < x1 < x2 < x3 < x4 < < xn2 < xn1 < xn b. (x1, y1) ͓ ͔ ͓ ͔ ͓ ͔ x0, x2 x2, x4 xn2, xn ͓ ͔ On each (double) subinterval xi2, xi , you can approximate f by a polynomial p of (x0, y0) degree less than or equal to 2. (See Exercise 47.) For example, on the subinterval ͓ ͔ ͑ ͒ x0, x2 , choose the polynomial of least degree passing through the points x0, y0 , ͑ ͒ ͑ ͒ x x1, y1 , and x2, y2 , as shown in Figure 4.45. Now, using p as an approximation of f on x x x x 0 1 2 n this subinterval, you have, by Theorem 4.18,

x2 x2

x2 x2 ͵ f ͑x͒ dx Ϸ ͵ p͑x͒ dx ͵ ͑ ͒ Ϸ ͵ ͑ ͒ x x p x dx f x dx 0 0 x x 0 0 x x x x 2 0 ΄p͑x ͒ 4p΂ 0 2΃ p͑x ͒΅ Figure 4.45 6 0 2 2 2͓͑b a͒͞n͔ ͓ p͑x ͒ 4p ͑x ͒ p͑x ͔͒ 6 0 1 2 b a ͓ f ͑x ͒ 4 f ͑x ͒ f ͑x ͔͒. 3n 0 1 2 Repeating this procedure on the entire interval ͓a, b͔ produces the next theorem.

THEOREM 4.19 Simpson’s Rule REMARK Observe that the Let f be continuous on ͓a, b͔ and let n be an even integer. Simpson’s Rule for ͐b ͑ ͒ coefficients in Simpson’s Rule approximating a f x dx is have the following pattern. b ͵ ͑ ͒ Ϸ b a ͓ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ . . . f x dx f x0 4 f x1 2 f x2 4 f x3 142424. . .4241 a 3n

4 f ͑xn1͒ f ͑xn͔͒. → ͐b ͑ ͒ Moreover, as n , the right-hand side approaches a f x dx.

͐ In Example 1, the Trapezoidal Rule was used to estimate 0 sin x dx. In the next REMARK In Section 4.2, example, Simpson’s Rule is applied to the same integral. Example 8, the Midpoint Rule with n 4 approximates Approximation with Simpson’s Rule ͵ sin x dx as 2.052. In 0 See LarsonCalculus.com for an interactive version of this type of example. Example 1, the Trapezoidal Rule with n 4 gives an Use Simpson’s Rule to approximate approximation of 1.896. In Example 2, Simpson’s Rule ͵ sin x dx. 0 with n 4 gives an approximation of 2.005. The Compare the results for n 4 and n 8. antiderivative would produce Solution When n 4, you have the true value of 2. 3 ͵ sin x dx Ϸ ΂sin 0 4 sin 2 sin 4 sin sin ΃ Ϸ 2.005. 0 12 4 2 4 When n 8, you have ͵ sin x dx Ϸ 2.0003. 0

FOR FURTHER INFORMATION Error Analysis For proofs of the formulas used for When you use an approximation technique, it is important to know how accurate you estimating the errors involved in can expect the approximation to be. The next theorem, which is listed without proof, the use of the Midpoint Rule and gives the formulas for estimating the errors involved in the use of Simpson’s Rule and Simpson’s Rule, see the article the Trapezoidal Rule. In general, when using an approximation, you can think of the “Elementary Proofs of Error error E as the difference between ͐b f͑x͒ dx and the approximation. Estimates for the Midpoint and a Simpson’s Rules” by Edward C. Fazekas, Jr. and Peter R. Mercer in THEOREM 4.20 Errors in the Trapezoidal Rule and Simpson’s Rule Mathematics Magazine. To view If f has a continuous second derivative on ͓a, b͔, then the error E in this article, go to MathArticles.com. ͐b ͑ ͒ approximating a f x dx by the Trapezoidal Rule is ͑b a͒3 ԽEԽ ͓max Խ f ͑x͒Խ͔, a x b. Trapezoidal Rule 12n2 Moreover, if f has a continuous fourth derivative on ͓a, b͔, then the error E ͐b ͑ ͒ in approximating a f x dx by Simpson’s Rule is 5 ͑b a͒ ͑ ͒ ԽEԽ ͓max Խ f 4 ͑x͒Խ ͔, a x b. Simpson’s Rule 180n4

TECHNOLOGY If you have Theorem 4.20 states that the errors generated by the Trapezoidal Rule and ͑ ͒ access to a computer algebra Simpson’s Rule have upper bounds dependent on the extreme values of f͑x͒ and f 4 ͑x͒ system, use it to evaluate the in the interval ͓a, b͔. Furthermore, these errors can be made arbitrarily small by ͑ ͒ definite integral in Example 3. increasing n, provided that f and f 4 are continuous and therefore bounded in ͓a, b͔. You should obtain a value of 1 The Approximate Error in the Trapezoidal Rule ͵ Ί1 x2 dx 0 Determine a value of n such that the Trapezoidal Rule will approximate the value of 1 ͓Ί2 ln ͑1 Ί2͔͒ 1 2 ͵ Ί1 x2 dx Ϸ 1.14779. 0 with an error that is less than or equal to 0.01. (The symbol “ln” represents the natural logarithmic function, Solution Begin by letting f ͑x͒ Ί1 x2 and finding the second derivative of f. which you will study in f͑x͒ x͑1 x2͒1͞2 and f͑x͒ ͑1 x2͒3͞2 Section 5.1.) The maximum value of Խ f ͑x͒Խ on the interval ͓0, 1͔ is Խ f ͑0͒Խ 1. So, by Theorem 4.20, you can write y ͑b a͒3 1 1 ԽEԽ Խ f ͑0͒Խ ͑1͒ . 12n 2 12n 2 12n 2 2 To obtain an error E that is less than 0.01, you must choose n such that y = 1 + x2 1͑͞12n2͒ 1͞100. 2 Ί100 Ϸ 100 12n n 12 2.89 1 So, you can choose n 3 (because n must be greater than or equal to 2.89) and apply n = 3 the Trapezoidal Rule, as shown in Figure 4.46, to obtain 1 1 2 2 ͵ Ί 2 Ϸ ͓Ί 2 Ί ͑1͒ Ί ͑2͒ Ί 2͔ x 1 x dx 1 0 2 1 3 2 1 3 1 1 1 2 0 6 Ϸ 1.154. 1 So, by adding and subtracting the error from this estimate, you know that 1.144 ͵ Ί1 x2 dx 1.164 0 1 Figure 4.46 1.144 ͵ Ί1 x2 dx 1.164. 0

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

Using the Trapezoidal Rule and Simpson’s Rule In Estimating Errors In Exercises 23–26, use the error Exercises 1–10, use the Trapezoidal Rule and Simpson’s Rule formulas in Theorem 4.20 to estimate the errors in approximating using (a) the Trapezoidal Rule and ,4 ؍ to approximate the value of the definite integral for the given the integral, with n value of n. Round your answer to four decimal places and (b) Simpson’s Rule. compare the results with the exact value of the definite integral. 3 5 ͵ 3 ͵ ͑ ͒ 2 2 23.2x dx 24. 5x 2 dx x2 1.͵ x2 dx, n 4 2. ͵ ΂ 1΃ dx, n 4 1 3 4 0 1 ͵4 1 ͵ 2 3 25. dx 26. cos x dx 2 ͑x 1͒2 ͵ 3 ͵ 2 0 3. x dx, n 4 4. 2 dx, n 4 0 2 x 3 8 Estimating Errors In Exercises 27–30, use the error 5.͵ x3 dx, n 6 6. ͵ Ί3 x dx, n 8 formulas in Theorem 4.20 to find n such that the error in the 1 0 approximation of the definite integral is less than or equal to 9 4 0.00001 using (a) the Trapezoidal Rule and (b) Simpson’s Rule. ͵ Ί ͵ ͑ 2͒ 7. x dx, n 8 8. 4 x dx, n 6 3 1 4 1 1 1 27.͵ dx 28. ͵ dx 1 2 1 x 0 1 x 9. ͵ dx, n 4 ͑ ͒2 2 ͞2 0 x 2 29.͵ Ίx 2 dx 30. ͵ sin x dx 2 0 0 10. ͵ xΊx2 1 dx, n 4 0 Estimating Errors Using Technology In Exercises 31–34, Using the Trapezoidal Rule and Simpson’s Rule In use a computer algebra system and the error formulas to find Exercises 11–20, approximate the definite integral using the n such that the error in the approximation of the definite Compare integral is less than or equal to 0.00001 using (a) the .4 ؍ Trapezoidal Rule and Simpson’s Rule with n these results with the approximation of the integral using a Trapezoidal Rule and (b) Simpson’s Rule. graphing utility. 2 2 31.͵ Ί1 x dx 32. ͵ ͑x 1͒2͞3 dx 2 2 1 0 0 11.͵ Ί1 x3 dx 12. ͵ dx 3 1 1 0 0 Ί1 x 33.͵ tan x2 dx 34. ͵ sin x2 dx 1 0 0 13.͵ Ίx Ί1 x dx 14. ͵ Ίx sin x dx 0 ͞2 35. Finding the Area of a Region Approximate the area of Ί͞2 Ί͞4 the shaded region using 15.͵ sin x2 dx 16. ͵ tan x2 dx 0 0 (a) the Trapezoidal Rule with n 4. 3.1 ͞2 (b) Simpson’s Rule with n 4. 17.͵ cos x2 dx 18. ͵ Ί1 sin 2 x dx 3 0 y y ͞4 10 10 19. ͵ x tan x dx 0 8 8 sin x , x > 0 6 6 20. ͵ f͑x͒ dx, f ͑x͒ Ά x 0 1, x 0 4 4 2 2 WRITING ABOUT CONCEPTS x x 12345 246810 21. Polynomial Approximations The Trapezoidal Rule and Simpson’s Rule yield approximations of a definite Figure for 35 Figure for 36 integral ͐b f ͑x͒ dx based on polynomial approximations of a 36. Finding the Area of a Region Approximate the area of f. What is the degree of the polynomials used for each? the shaded region using 22. Describing an Error Describe the size of the error (a) the Trapezoidal Rule with n 8. when the Trapezoidal Rule is used to approximate ͐b ͑ ͒ ͑ ͒ (b) Simpson’s Rule with n 8. a f x dx when f x is a linear function. Use a graph to explain your answer. 37. Area Use Simpson’s Rule with n 14 to approximate the area of the region bounded by the graphs of y Ίx cos x, y 0, x 0, and x ͞2.

38. Circumference The elliptic integral 41. Work To determine the size of the motor required to

͞2 operate a press, a company must know the amount of work Ί ͵ Ί 2 2 done when the press moves an object linearly 5 feet. The 8 3 1 3 sin d 0 variable force to move the object is gives the circumference of an ellipse. Use Simpson’s Rule F͑x͒ 100xΊ125 x3 with n 8 to approximate the circumference. where F is given in pounds and x gives the position of the unit 39. Surveying in feet. Use Simpson’s Rule with n 12 to approximate the Use the Trapezoidal Rule work W (in foot-pounds) done through one cycle when

to estimate the number 5 of square meters of land, ͵ ͑ ͒ W F x dx. where x and y are meas- 0 ured in meters, as shown 42. Approximating a Function The table lists several in the figure. The land is measurements gathered in an experiment to approximate an bounded by a stream and unknown continuous function y f ͑x͒. two straight roads that meet at right angles. x 0.00 0.25 0.50 0.75 1.00

x 0 100 200 300 400 500 y 4.32 4.36 4.58 5.79 6.14 y 125 125 120 112 90 90 x 1.25 1.50 1.75 2.00

x 600 700 800 900 1000 y 7.25 7.64 8.08 8.14 y 95 88 75 35 0 (a) Approximate the integral

y 2 ͵ f ͑x͒ dx 150 Road 0 Stream using the Trapezoidal Rule and Simpson’s Rule. 100 (b) Use a graphing utility to find a model of the form y ax3 bx2 cx d for the data. Integrate the ͓ ͔ 50 Road resulting polynomial over 0, 2 and compare the result with the integral from part (a). x 200 400 600 800 1000 Approximation of Pi In Exercises 43 and 44, use to approximate ␲ using the given 6 ؍ Simpson’s Rule with n equation. (In Section 5.7, you will be able to evaluate the integral using inverse trigonometric functions.)

͑ ͒ 1͞2 1 40. HOW DO YOU SEE IT? The function f x is 6 4 concave upward on the interval ͓0, 2͔ and the func- 43. ͵ dx 44. ͵ dx Ί1 x2 1 x2 tion g͑x͒ is concave downward on the interval ͓0, 2͔. 0 0

y y 45. Using Simpson’s Rule Use Simpson’s Rule with n 10 and a computer algebra system to approximate t in the 5 f(x) 5 integral equation g(x) 4 4 t 3 3 ͵ sin Ίx dx 2. 0 2 2 46. Proof Prove that Simpson’s Rule is exact when approx- 1 imating the integral of a cubic polynomial function, and x x demonstrate the result with n 4 for 12 12 1 (a) Using the Trapezoidal Rule with n 4 , which ͵ x3 dx. integral would be overestimated? Which integral 0 would be underestimated? Explain your reasoning. 47. Proof Prove that you can find a polynomial (b) Which rule would you use for more accurate ͐2 ͑ ͒ ͐2 ͑ ͒ p͑x͒ Ax2 Bx C approximations of 0 f x dx and 0 g x dx, the Trapezoidal Rule or Simpson’s Rule? Explain your ͑ ͒ ͑ ͒ that passes through any three points x1, y1 , x2, y2 , and reasoning. ͑ ͒ x3, y3 , where the xi ’s are distinct. Henryk Sadura/Shutterstock.com

Section 4.6 (page 310) Trapezoidal Simpson’s Exact 1. 2.7500 2.6667 2.6667 3. 4.2500 4.0000 4.0000 5. 20.2222 20.0000 20.0000 7. 12.6640 12.6667 12.6667 9. 0.3352 0.3334 0.3333 Trapezoidal Simpson’s Graphing Utility 11. 3.2833 3.2396 3.2413 13. 0.3415 0.3720 0.3927 15. 0.5495 0.5483 0.5493 17. 0.0975 0.0977 0.0977 19. 0.1940 0.1860 0.1858 21. Trapezoidal: Linear (1st-degree) polynomials Simpson’s: Quadratic (2nd-degree) polynomials 1 1 23. (a) 1.500 (b) 0.000 25. (a)4 (b) 12 27. (a) n 366 (b) n 26 29. (a)n 77 (b) n 8 31. (a) n 130 (b) n 12 33. (a) n 643 (b) n 48 35. (a) 24.5 (b) 25.67 37. 0.701 39. 89,250 m2 41. 10,233.58 ft-lb 43. 3.1416 45. 2.477 47. Proof