4.3 Riemann Sums and Definite Integrals

266 Chapter 4 Integration

Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals.

Riemann Sums In the definition of area given in Section 4.2, the partitions have subintervals of equal width. This was done only for computational convenience. The next example shows that it is not necessary to have subintervals of equal width.

A Partition with Subintervals of Unequal Widths

y f(x) = x Consider the region bounded by the graph of 1 f ͑x͒ Ίx n − 1 n and the x- axis for 0 x 1, as shown in Figure 4.18. Evaluate the limit

n . . . lim f ͑c ͒ x → ͚ i i 2 n n i 1 2͞ 2 1 where ci is the right endpoint of the partition given by ci i n and xi is the width x n of the i th interval. 2 (n − 1)2 1 2 . . . 1 n2 n2 n2 Solution The width of the i th interval is The subintervals do not have equal i 2 ͑i 1͒2 x widths. i n2 n2 Figure 4.18 i 2 i 2 2i 1 n2 2i 1 . n2 So, the limit is n n i 2 2i 1 lim f ͑c ͒ x lim Ί ΂ ΃ → ͚ i i → ͚ 2 2 n i1 n i1 n n 1 n lim ͑2i 2 i͒ → 3 ͚ n n i1 y 1 n͑n 1͒͑2n 1͒ n͑n 1͒ lim ΄2΂ ΃ ΅ n→ n3 6 2 x = y2 4n3 3n2 n 1 (1, 1) lim Area = 1 n→ 6n3 3 2 . 3

(0, 0) x 1 From Example 7 in Section 4.2, you know that the region shown in Figure 4.19 has 1 an area of 3. Because the square bounded by 0 x 1 and 0 y 1 has an area of 1, you can conclude that the area of the region shown in Figure 4.18 has an area of 2 The area of the region bounded by 3. This agrees with the limit found in Example 1, even though that example used a the graph of x y2 and the y -axis partition having subintervals of unequal widths. The reason this particular partition gave 1 for 0 y 1 is 3. the proper area is that as n increases, the width of the largest subinterval approaches Figure 4.19 zero. This is a key feature of the development of definite integrals.

In Section 4.2, the limit of a sum was used to define the area of a region in the plane. Finding area by this means is only one of many applications involving the limit of a sum. A similar approach can be used to determine quantities as diverse as arc lengths, average values, centroids, volumes, work, and surface areas. The next definition is named after Georg Friedrich Bernhard Riemann. Although the definite integral had been defined and used long before Riemann’s time, he generalized the concept to cover a broader category of functions. In the definition of a Riemann sum below, note that the function f has no restrictions other than being defined on the interval ͓a, b͔. (In Section 4.2, the function f was assumed to be continuous and nonnegative because you were finding the area under a curve.)

Definition of Riemann Sum Let f be defined on the closed interval ͓a, b͔, and let be a partition of ͓a, b͔ GEORG FRIEDRICH BERNHARD given by RIEMANN (1826-1866) a x0 < x1 < x2 < . . . < xn1 < xn b German mathematician Riemann did his most famous work in the where xi is the width of the i th subinterval areas of non-Euclidean geometry, ͓ ͔ differential equations, and number xi1, xi . ith subinterval theory. It was Riemann’s results If c is any point in the i th subinterval, then the sum in physics and mathematics that i formed the structure on which n ͑ ͒ Einstein’s General Theory of ͚ f ci xi , xi1 ci xi Relativity is based. i1 See LarsonCalculus.com to read is called a Riemann sum of f for the partition . (The sums in Section 4.2 are more of this biography. examples of Riemann sums, but there are more general Riemann sums than those covered there.)

The width of the largest subinterval of a partition is the norm of the partition and is denoted by ʈʈ. If every subinterval is of equal width, then the partition is regular and the norm is denoted by

b a ʈʈ x . Regular partition n

For a general partition, the norm is related to the number of subintervals of ͓a, b͔ in the following way. b a ʈʈ n General partition

So, the number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is,ʈʈ → 0 implies that n → . The converse of this statement is not true. For example, let n be the partition of the interval ͓0, 1͔ given by

1 1 . . . 1 1 1 0 < n < n1 < < < < < 1. ⏐⏐ Δ⏐⏐ = 1 2 2 8 4 2 2 1 As shown in Figure 4.20, for any positive value of n, the norm of the partition n is 2. ʈʈ 0 1 1 1 1 So, letting n approach infinity does not force to approach 0. In a regular partition, 1 8 4 2 however, the statements 2n ʈʈ → 0 and n → n → does not imply that ʈʈ → 0. Figure 4.20 are equivalent. INTERFOTO/Alamy

For insight into the history of the n definite integral, see the article lim f ͑c ͒ x L. ʈʈ→ ͚ i i “The Evolution of Integration” by 0 i1 A. Shenitzer and J. Steprans in The To say that this limit exists means there exists a real number L such that for each > 0, American Mathematical Monthly. there exists a > 0 such that for every partition with ʈʈ < , it follows that To view this article, go to n MathArticles.com. L f ͑c ͒ x Խ ͚ i iԽ < i1 regardless of the choice of ci in the i th subinterval of each partition .

Definition of Definite Integral If f is defined on the closed interval ͓a, b͔ and the limit of Riemann sums over partitions

n lim f ͑c ͒ x ʈʈ→ ͚ i i 0 i1 exists (as described above), then f is said to be integrable on ͓a, b͔ and the limit is denoted by

n b lim f ͑c ͒ x ͵ f ͑x͒ dx. ʈʈ→ ͚ i i 0 i1 a REMARK Later in this chapter, you will learn The limit is called the definite integral of f from a to b. The number a is the convenient methods for lower limit of integration, and the number b is the upper limit of integration. ͐b ͑ ͒ calculating a f x dx for continuous functions. For now, you must use the limit definition. It is not a coincidence that the notation for definite integrals is similar to that used for indefinite integrals. You will see why in the next section when the Fundamental Theorem of Calculus is introduced. For now, it is important to see that definite integrals and indefinite integrals are different concepts. A definite integral is a number, whereas an indefinite integral is a family of functions. Though Riemann sums were defined for functions with very few restrictions, a sufficient condition for a function f to be integrable on ͓a, b͔ is that it is continuous on ͓a, b͔. A proof of this theorem is beyond the scope of this text.

THEOREM 4.4 Continuity Implies Integrability If a function f is continuous on the closed interval ͓a, b͔, then f is integrable ͓ ͔ ͐b ͑ ͒ on a, b . That is,a f x dx exists.

Exploration The Converse of Theorem 4.4 Is the converse of Theorem 4.4 true? That is, when a function is integrable, does it have to be continuous? Explain your reasoning and give examples. Describe the relationships among continuity, differentiability, and integrability. Which is the strongest condition? Which is the weakest? Which conditions imply other conditions?

Evaluating a Definite Integral as a Limit

1 Evaluate the definite integral ͵ 2x dx. 2 Solution The function f ͑x͒ 2x is integrable on the interval ͓2, 1͔ because it is continuous on ͓2, 1͔. Moreover, the definition of integrability implies that any partition whose norm approaches 0 can be used to determine the limit. For y computational convenience, define by subdividing ͓2, 1͔ into n subintervals of equal width 2 b a 3 x x . i n n 1 Choosing c as the right endpoint of each subinterval produces f(x) = 2x i ͑ ͒ 3i x ci a i x 2 . 1 n So, the definite integral is

1 n ͵ 2x dx lim f ͑c ͒ x ʈʈ→ ͚ i i 2 0 i1 −2 n lim f ͑c ͒ x → ͚ i n i1 −3 n 3i 3 lim 2΂2 ΃΂ ΃ → ͚ n i1 n n −4 6 n 3i lim ΂2 ΃ → ͚ n n i1 n Because the definite integral is 6 n 3 n lim ΂2 1 i΃ negative, it does not represent the → ͚ ͚ n n i1 ni1 area of the region. 6 3 n͑n 1͒ Figure 4.21 lim Ά2n ΄ ΅· n→ n n 2 9 lim ΂12 9 ΃ n→ n 3.

Because the definite integral in Example 2 is negative, it does not represent the area y of the region shown in Figure 4.21. Definite integrals can be positive, negative, or zero. For a definite integral to be interpreted as an area (as defined in Section 4.2), the f function f must be continuous and nonnegative on ͓a, b͔, as stated in the next theorem. The proof of this theorem is straightforward—you simply use the definition of area given in Section 4.2, because it is a Riemann sum.

THEOREM 4.5 The Definite Integral as the Area of a Region x If f is continuous and nonnegative on the closed interval ͓a, b͔, then the area ab of the region bounded by the graph of f, the x -axis, and the vertical lines You can use a definite integral to find x a and x b is the area of the region bounded by the b graph of f, the x -axis,x a, and Area ͵ f ͑x͒ dx. x b. a Figure 4.22 (See Figure 4.22.)

y f(x) = 4x − x2 As an example of Theorem 4.5, consider the region bounded by the graph of ͑ ͒ 2 4 f x 4x x and the x- axis, as shown in Figure 4.23. Because f is continuous and nonnegative on the 3 closed interval ͓0, 4͔, the area of the region is 4 2 Area ͵ ͑4x x2͒ dx. 0 1 A straightforward technique for evaluating a definite integral such as this will be discussed in Section 4.4. For now, however, you can evaluate a definite integral in two x ways—you can use the limit definition or you can check to see whether the definite 1234 integral represents the area of a common geometric region, such as a rectangle, triangle, 4 or semicircle. Area ͵ ͑4x x2͒ dx 0 Figure 4.23 Areas of Common Geometric Figures

Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula. 3 3 2 a.͵ 4 dx b.͵ ͑x 2͒ dx c. ͵ Ί4 x2 dx 1 0 2 Solution A sketch of each region is shown in Figure 4.24. a. This region is a rectangle of height 4 and width 2.

3 ͵ 4 dx (Area of rectangle) 4͑2͒ 8 1 b. This region is a trapezoid with an altitude of 3 and parallel bases of lengths 2 and 1 ͑ ͒ 5. The formula for the area of a trapezoid is 2h b1 b2 . 3 1 21 ͵ ͑x 2͒ dx (Area of trapezoid) ͑3͒͑2 5͒ 0 2 2 c. This region is a semicircle of radius 2. The formula for the area of a semicircle is 1 2 2 r .

2 1 ͵ Ί4 x2 dx (Area of semicircle) ͑22͒ 2 2 2

y y y f(x) = 4 f(x) = x + 2 4 5 4 − 2 4 f(x) = 4 x 3 3 3 2 2 1 1 1 x x x 1234 12345 −2 −112 (a) (b) (c) Figure 4.24

The variable of integration in a definite integral is sometimes called a dummy variable because it can be replaced by any other variable without changing the value of the integral. For instance, the definite integrals 3 3 ͵ ͑x 2͒ dx and ͵ ͑t 2͒ dt 0 0 have the same value.

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.3 Riemann Sums and Definite Integrals 271 Properties of Definite Integrals The definition of the definite integral of f on the interval ͓a, b͔ specifies that a < b. Now, however, it is convenient to extend the definition to cover cases in which a b or a > b. Geometrically, the next two definitions seem reasonable. For instance, it makes sense to define the area of a region of zero width and finite height to be 0.

Definitions of Two Special Definite Integrals a 1. If f is defined at x a, then ͵ f ͑x͒ dx 0. a a b 2. If f is integrable on ͓a, b͔, then ͵ f ͑x͒ dx ͵ f ͑x͒ dx. b a

Evaluating Definite Integrals

See LarsonCalculus.com for an interactive version of this type of example. Evaluate each definite integral. 0 a.͵ sin x dx b. ͵ ͑x 2͒ dx 3 Solution a. Because the sine function is defined at x , and the upper and lower limits of integration are equal, you can write ͵ sin x dx 0. ͐0͑ ͒ b. The integral 3 x 2 dx is the same as that given in Example 3(b) except that the upper and lower limits are interchanged. Because the integral in Example 3(b) has a 21 value of 2 , you can write 0 3 21 ͵ ͑x 2͒ dx ͵ ͑x 2͒ dx . 3 0 2

y b ∫ f(x) dx In Figure 4.25, the larger region can be divided at x c into two subregions whose a intersection is a line segment. Because the line segment has zero area, it follows that the area of the larger region is equal to the sum of the areas of the two smaller regions. f

THEOREM 4.6 Additive Interval Property If f is integrable on the three closed intervals determined by a, b, and c, then b c b ͵ ͑ ͒ ͵ ͑ ͒ ͵ ͑ ͒ x f x dx f x dx f x dx. acb a a c

c b ∫ f(x) dx+ ∫ f(x) dx a c Figure 4.25 Using the Additive Interval Property

1 0 1 ͵ ͵ ͵ Theorem 4.6 ԽxԽ dx x dx x dx 1 1 0 1 1 Area of a triangle 2 2 1

Because the definite integral is defined as the limit of a sum, it inherits the properties of summation given at the top of page 255.

THEOREM 4.7 Properties of Definite Integrals If f and g are integrable on ͓a, b͔ and k is a constant, then the functions kf and f ± g are integrable on ͓a, b͔, and b b 1. ͵ kf ͑x͒ dx k͵ f ͑x͒ dx a a

b b b 2. ͵ ΄f ͑x͒ ± g͑x͒] dx ͵ f ͑x͒ dx ± ͵ g͑x͒ dx. a a a REMARK Property 2 of Theorem 4.7 can be extended to cover any finite number of functions (see Example 6). Evaluation of a Definite Integral

3 Evaluate ͵ ͑x2 4x 3͒ dx using each of the following values. 1 3 26 3 3 ͵ x2 dx , ͵ x dx 4, ͵ dx 2 1 3 1 1 Solution 3 3 3 3 ͵ ͑x2 4x 3͒ dx ͵ ͑x2͒ dx ͵ 4x dx ͵ ͑3͒ dx 1 1 1 1 3 3 3 ͵ x2 dx 4͵ x dx 3͵ dx 1 1 1 26 ΂ ΃ 4͑4͒ 3͑2͒ 3 4 3

y If f and g are continuous on the closed interval ͓a, b͔ and 0 f ͑x͒ g͑x͒ for a x b, then the following properties are true. First, the area of the region bounded by the graph of f and the x- axis (between a and b ) must be nonnegative. Second, this g area must be less than or equal to the area of the region bounded by the graph of g and the x- axis (between a and b ), as shown in Figure 4.26. These two properties are f generalized in Theorem 4.8.

THEOREM 4.8 Preservation of Inequality 1. If f is integrable and nonnegative on the closed interval ͓a, b͔, then x b ab 0 ͵ f ͑x͒ dx. b b a ͵ f ͑x͒ dx ͵ g͑x͒ dx ͓ ͔ ͑ ͒ ͑ ͒ a a 2. If f and g are integrable on the closed interval a, b and f x g x for Figure 4.26 every x in ͓a, b͔, then b b ͵ f ͑x͒ dx ͵ g͑x͒ dx. a a A proof of this theorem is given in Appendix A. See LarsonCalculus.com for Bruce Edwards’s video of this proof.

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

Evaluating a Limit In Exercises 1 and 2, use Example 1 as 15.f ͑x͒ 4 ԽxԽ 16. f ͑x͒ x2 a model to evaluate the limit y y n lim fͧc ͨ x 8 4 ͚ i i ؕ→ 1؍n i 6 3 over the region bounded by the graphs of the equations. 4 2 1. f͑x͒ Ίx, y 0, x 0, x 3 2 1 3i2 x x ΂Hint: Let ci 2 .΃ n −424−2 −1 123 2. f͑x͒ Ί3 x, y 0, x 0, x 1 4 i3 17.f ͑x͒ 25 x2 18. f ͑x͒ ΂Hint: Let c .΃ x2 2 i n3 y y Evaluating a Definite Integral as a Limit In Exercises 3–8, evaluate the definite integral by the limit definition. 6 3 15 ͵ ͵ 1 3. 8 dx 4. x dx 10 2 2 1 4 5 ͵ 3 ͵ 2 x x 5. x dx 6. 4x dx − − − 1 1 64 2 246 −11 2 1 ͑ 2 ͒ ͑ 2 ͒ 7.͵ x 1 dx 8. ͵ 2x 3 dx 19.f ͑x͒ cos x 20. f ͑x͒ tan x 1 2 y y Writing a Limit as a Definite Integral In Exercises 9–12, write the limit as a definite integral on the interval [a, b], where

ci is any point in the i th subinterval. 1 1 Limit Interval n ͑ ͒ ͓ ͔ 9. lim ͚ 3ci 10 xi 1, 5 ʈʈ→0 i1 x x n ππ ππ ͑ ͒2 ͓ ͔ 42 42 10. lim ͚ 6ci 4 ci xi 0, 4 ʈʈ→0 i1 ͑ ͒ 3 ͑ ͒ ͑ ͒2 n 21.g y y 22. f y y 2 Ί 2 ͓ ͔ 11. lim ͚ ci 4 xi 0, 3 y y ʈʈ→0 i1 n 3 ͓ ͔ 4 4 12. lim ͚ ΂ ΃ xi 1, 3 ʈʈ→ 2 0 i1 ci 3 3 Writing a Definite Integral In Exercises 13–22, set up a 2 2 definite integral that yields the area of the region. (Do not 1 1 evaluate the integral.) x x 13.f ͑x͒ 5 14. f ͑x͒ 6 3x 2684 1342 y y Evaluating a Definite Integral Using a Geometric 5 6 Formula In Exercises 23–32, sketch the region whose area is 4 5 given by the definite integral. Then use a geometric formula to 4 ͧ ͨ 3 evaluate the integral a > 0, r > 0 . 3 3 6 2 2 23.͵ 4 dx 24. ͵ 6 dx 1 1 0 4 x x 4 8 −2 −1 12345 x 12345 25.͵ x dx 26. ͵ dx 0 0 4

2 3 44. Using Properties of Definite Integrals Given 27.͵ ͑3x 4͒ dx 28. ͵ ͑8 2x͒ dx 0 0 1 1 1 a ͵ ͑ ͒ ͵ ͑ ͒ f x dx 0 and f x dx 5 29.͵ ͑1 ԽxԽ͒ dx 30. ͵ ͑a ԽxԽ͒ dx 1 0 1 a 7 r evaluate 31.͵ Ί49 x2 dx 32. ͵ Ίr 2 x2 dx 0 1 0 7 r (a)͵ f ͑x͒ dx. (b) ͵ f ͑x͒ dx ͵ f ͑x͒ dx. 1 0 1 Using Properties of Definite Integrals In Exercises 1 1 ͑ ͒ ͑ ͒ 33–40, evaluate the integral using the following values. (c)͵ 3f x dx. (d) ͵ 3f x dx. 1 0 4 4 4 45. Estimating a Definite Integral Use the table of values 2 ؍ dx ͵ ,6 ؍ x dx ͵ ,60 ؍ x 3 dx ͵ to find lower and upper estimates of 2 2 2 2 2 10 ͵ ͵ 3 ͵ f͑x͒ dx. 33. x dx 34. x dx 4 2 0 4 4 ͵ ͵ Assume that f is a decreasing function. 35. 8x dx 36. 25 dx 2 2 4 4 x 024 6 8 10 ͵ ͑ ͒ ͵ ͑ 3 ͒ 37. x 9 dx 38. x 4 dx 2 2 ͑ ͒ 4 4 f x 32 24 12 4 20 36 ͵ ͑1 3 ͒ ͵ ͑ 3͒ 39. 2 x 3x 2 dx 40. 10 4x 3x dx 2 2 46. Estimating a Definite Integral Use the table of values 41. Using Properties of Definite Integrals Given to estimate 6 5 7 ͵ ͑ ͒ ͵ f ͑x͒ dx 10 and ͵ f ͑x͒ dx 3 f x dx. 0 0 5 evaluate Use three equal subintervals and the (a) left endpoints, 7 0 (b) right endpoints, and (c) midpoints. When f is an increasing (a)͵ f ͑x͒ dx. (b) ͵ f ͑x͒ dx. function, how does each estimate compare with the actual 0 5 value? Explain your reasoning. 5 5 (c)͵ f ͑x͒ dx. (d) ͵ 3f ͑x͒ dx. 5 0 x 0123456 42. Using Properties of Definite Integrals Given f ͑x͒ 6 0818305080 3 6 ͵ f ͑x͒ dx 4 and ͵ f͑x) dx 1 0 3 47. Think About It The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite evaluate integral by using geometric formulas. 6 3 (a)͵ f ͑x͒ dx. (b) ͵ f ͑x͒ dx. y 0 6 (4, 2) 3 6 2 (c)͵ f ͑x͒ dx. (d) ͵ 5f ͑x͒ dx. 3 3 1 f 43. Using Properties of Definite Integrals Given x −4 −113456 6 6 −1 ͵ ͑ ͒ ͵ ͑ ͒ − − f x dx 10 and g x dx 2 ( 4, 1) 2 2 evaluate 6 6 2 6 (a)͵ ͓ f ͑x͒ g͑x͔͒ dx. (b) ͵ ͓g͑x͒ f ͑x͔͒ dx. (a)͵ f͑x͒ dx (b) ͵ f͑x͒ dx 2 2 0 2 6 6 2 6 (c)͵ 2g͑x͒ dx. (d) ͵ 3f ͑x͒ dx. (c)͵ f͑x͒ dx (d) ͵ f͑x͒ dx 2 2 4 4 6 6 (e)͵ Խf͑x͒Խ dx (f) ͵ ͓ f͑x͒ 2͔ dx 4 4

48. Think About It The graph of f consists of line segments, 51. Think About It A function f is defined below. Use ͐8 ͑ ͒ as shown in the figure. Evaluate each definite integral by using geometric formulas to find 0 f x dx. geometric formulas. 4, x < 4 f͑x͒ y Άx, x 4 4 3 52. Think About It A function f is defined below. Use (3, 2) (4, 2) ͐12 ͑ ͒ 2 (11, 1) geometric formulas to find 0 f x dx. 1 f x 6, x > 6 − f͑x͒ 1 123456 8 1011 Ά 1 x 9, x 6 −2 2 −3 (8, −2) −4 WRITING ABOUT CONCEPTS 1 4 Approximation In Exercises 53–56, determine which (a)͵ f͑x͒ dx (b) ͵ 3 f͑x͒ dx 0 3 value best approximates the definite integral. Make your 7 11 selection on the basis of a sketch. ͵ ͑ ͒ ͵ ͑ ͒ (c)f x dx (d) f x dx 4 0 5 ͵ Ί 11 10 53. x dx (e)͵ f͑x͒ dx (f) ͵ f ͑x͒ dx 0 0 4 (a) 5 (b)3 (c) 10 (d) 2 (e) 8 49. Think About It Consider the function f that is continuous 1͞2 on the interval ͓5, 5͔ and for which 54. ͵ 4 cos x dx 0 5 (a) 4 (b)4 (c) 16 (d)2 (e) 6 ͵ f͑x͒ dx 4. 3 1 0 55. ͵ 2 sin x dx Evaluate each integral. 0 1 5 5 3 (a) 6 (b)2 (c) 4 (d) 4 (a)͵ ͓ f͑x͒ 2͔ dx (b) ͵ f͑x 2͒ dx 9 0 2 56. ͵ ͑1 Ίx͒ dx 5 5 ͑ ͒ ͑ ͒ 0 (c)͵ f x dx (f is even.) (d)͵ f x dx (f is odd.) 5 5 (a)3 (b) 9 (c) 27 (d) 3

57. Determining Integrability Determine whether the 50. HOW DO YOU SEE IT? Use the figure to fill function in the blank with the symbol <, >, or . Explain 1 your reasoning. f͑x͒ x 4 y is integrable on the interval ͓3, 5͔. Explain. 6 5 58. Finding a Function Give an example of a function ͓ ͔ 4 that is integrable on the interval 1, 1 , but not continuous on ͓1, 1͔. 3 2 1 Finding Values In Exercises 59–62, find possible values of x 123456 a and b that make the statement true. If possible, use a graph to support your answer. (There may be more than one correct (a) The interval ͓1, 5͔ is partitioned into n subintervals answer.) x x of equal width , and i is the left endpoint of the 1 5 b ith subinterval. ͵ ͑ ͒ ͵ ͑ ͒ ͵ ͑ ͒ 59. f x dx f x dx f x dx 2 1 a n 5 ͑ ͒ ᭿ ͵ ͑ ͒ 3 6 b 6 ͚ f xi x f x dx ͵ ͑ ͒ ͵ ͑ ͒ ͵ ͑ ͒ ͵ ͑ ͒ i 1 1 60. f x dx f x dx f x dx f x dx 3 3 a 1 (b) The interval ͓1, 5͔ is partitioned into n subintervals b 61. ͵ sin x dx < 0 of equal width x, and xi is the right endpoint of the a ith subinterval. b 5 ͵ n 62. cos x dx 0 ͑ ͒ ᭿ ͵ ͑ ͒ a ͚ f xi x f x dx i1 1

True or False? In Exercises 63–68, determine whether the b 2 2 b a statement is true or false. If it is false, explain why or give an 71. Proof Prove that ͵ x dx . a 2 example that shows it is false. b b3 a3 72. Proof Prove that ͵ x2 dx . b b b 3 63. ͵ ͓ f͑x͒ g͑x͔͒ dx ͵ f͑x͒ dx ͵ g͑x͒ dx a a a a 73. Think About It Determine whether the Dirichlet function b b b 64. ͵ f͑x͒g͑x͒ dx ΄͵ f͑x͒ dx΅΄͵ g͑x͒ dx΅ 1, x is rational f͑x͒ a a a Ά0, x is irrational 65. If the norm of a partition approaches zero, then the number of subintervals approaches infinity. is integrable on the interval ͓0, 1͔. Explain. 66. If f is increasing on ͓a, b͔, then the minimum value of f͑x͒ on 74. Finding a Definite Integral The function ͓a, b͔ is f͑a͒. 0, x 0 67. The value of f ͑x͒ 1 Ά , 0 < x 1 b x ͵ f͑x͒ dx a is defined on ͓0, 1͔, as shown in the figure. Show that must be positive. 1 ͵ f͑x͒ dx 68. The value of 0 2 does not exist. Why doesn’t this contradict Theorem 4.4? ͵ sin ͑x2͒ dx 2 y

is 0. 5.0 4.0 69. Finding a Riemann Sum Find the Riemann sum for 3.0 f͑x͒ x2 3x over the interval ͓0, 8͔, where 2.0 x0 0, x1 1, x2 3, x3 7, and x4 8 1.0 x and where −0.5 0.5 1.0 1.5 2.0 c1 1, c2 2, c3 5, and c4 8. 75. Finding Values Find the constants a and b that maximize y the value of

100 b 2 80 ͵ ͑1 x ͒ dx. a 60 40 Explain your reasoning. 20 76. Step Function Evaluate, if possible, the integral

x 2 −2246810 ͵ ͠x͡ dx. 0 70. Finding a Riemann Sum Find the Riemann sum for 77. Using a Riemann Sum Determine f͑x͒ sin x over the interval ͓0, 2͔, where 1 lim ͓12 22 32 . . . n2͔ → 3 x 0, x , x , x , and x 2, n n 0 1 4 2 3 3 4 by using an appropriate Riemann sum. and where

2 3 PUTNAM EXAM CHALLENGE c , c , c , and c . 1 6 2 3 3 3 4 2 78. For each continuous function f: ͓0, 1͔ → ޒ, let

1 1 y I͑ f͒ ͵ x2 f͑x͒ dx and J͑x͒ ͵ x͑ f͑x͒͒2 dx. 1.5 0 0 1.0 Find the maximum value of I͑ f͒ J͑ f͒ over all such 0.5 functions f. x This problem was composed by the Committee on the Putnam Prize Competition. π 3π © The Mathematical Association of America. All rights reserved. 2 2

−1.5

Section 4.3 (page 273) Ί Ϸ 10 1. 2 3 3.464 3. 32 5. 0 7. 3 69. 272 71. Proof 5 3 4 73. No. No matter how small the subintervals, the number of both 9. ͵ ͑3x 10͒ dx 11. ͵ Ίx2 4 dx 13. ͵ 5 dx 1 0 0 rational and irrational numbers within each subinterval is 4 5 ͑ ͒ ͑ ͒ infinite, and f ci 0 or f ci 1. 15. ͵ ͑ x ͒ dx 17. ͵ ͑ x2͒ dx 1 4 Խ Խ 25 75. a 1 and b 1 maximize the integral. 77. 4 5 3 ͞2 2 19. ͵ cos x dx 21. ͵ y3 dy 0 0 23. y 25. y

3 Triangle

2 Rectangle 2

x x 1 234 5 2 4 A 12 A 8 27. y 29. y

1 Triangle 8 Trapezoid 4 x −1 1 x −1 123 −4

A 14 A 1 31. y 33.6 35. 48 37. 12 12 10 8 Semicircle 6 4 2 x −8−6 −4 −2 2468

−4

A 49͞2 39. 16 41. (a) 13 (b)10 (c) 0 (d) 30 43. (a) 8 (b)12 (c)4 (d) 30 45. 48, 88 47. (a) (b) 4 (c) ͑1 2͒ (d) 3 2 (e) 5 2 (f) 23 2 49. (a) 14 (b) 4 (c) 8 (d) 0 51. 40 53. a 55. d 57. No. There is a discontinuity at x 4. 59. a 2, b 5 61. Answers will vary. Sample answer: a , b 2 2 ͵ sin x dx < 0 y

x π 3π 2 2

−1

2 63. True 65. True 67. False. ͵ ͑x͒ dx 2 0