OpenStax Calculus Volume 1 – IASG Chapters 5–6 | Instructor Solutions & Study Guide 2026/2027

Chapter 5 Integration 5.1. Approximating Areas Section Exercises 1. State whether the given sums are equal or unequal. a. 10 i 1 i =  and 10 k 1 k =  b. 10 i 1 i =  and ( ) 15 6 5 i i =  − c. ( ) 10 1 1 i ii =  − and ( ) 9 0 1 j jj =  + d. ( ) 10 1 1 i ii =  − and ( ) 10 2 k 1 kk =  − Answer: a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting ji =−1 . d. They are equal; the first sum factors the terms of the second. In the following exercises, use the rules for sums of powers of integers to compute the sums. 2. i i=5 10 å Answer: 55 10 45 −= 3. i 2 i=5 10 å Answer: −= Suppose that 100 1 15 i i a =  = and 100 1 12 i i b =  =− . In the following exercises, compute the sums. 4. ai + b ( )i i=1 100 å Answer: 15 12 3 −= 5. ai - b ( )i i=1 100 å Answer: 15 12 27 − − = ( ) OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 6. 3ai - 4b ( )i i=1 100 å Answer: ( ) ( ) − − = 7. 5ai + 4b ( )i i=1 100 å Answer: ( ) ( ) + − = In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. 8. 100 k 2 ( ) -5k +1 k=1 20 å Answer: 20 20 20 2 1 1 1 100 5 1 18 ,004 0 k k k kk = = =  − + =      9. j 2 ( ) - 2 j j=1 50 å Answer: ( ) ( ) ( ) ( ) ( ) 50 50 2 11 51 2 jj 62 jj == − = − = 40,375 10. j 2 ( ) -10 j j=11 20 å Answer: 20 20 2 11 11 102 jj jj ==  − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21 11 69     = − − −         = 935 11. ( ) 2k 2 -100k é ëê ù ûú k=1 25 å Answer: 25 25 2 11 4 100 kk kk ==  − ( ) ( ) ( ) ( ) ( ) 4 25 26 51 50 25 26 6 =− =−10,400 OpenStax Calculus Volume 1 Instructor Answer and Solution Guide Let Ln denote the left-endpoint sum using n subintervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. 12. L4 for ( ) 1 1 fx x = − on   2, 3 Answer: 4 319 420 L = 13. R4 for g x( ) = cos( ) p x on   0, 1 Answer: 4 R =−0.25 14. L6 for ( ) ( ) 1 1 fx xx = − on   2, 5 Answer: 6 L = 0.5972 15. R6 for ( ) ( ) 1 1 fx xx = − on   2, 5 Answer: 6 R = 0.372 16. R4 for 2 1 x +1 on   −2, 2 Answer: 4 R = 2.20 17. L4 for 2 1 x +1 on   −2, 2 Answer: 4 L = 2.20 18. R8 for 2 xx −+ 21 on   0, 2 Answer: 8 R = 0.6875 19. L8 for 2 xx −+ 21 on   0, 2 Answer: 8 L = 0.6875 20. Compute the left and right Riemann sums—L4 and R4, respectively—for f x x ( ) =− ( ) 2 on   −2, 2 . Compute their average value and compare it with the area under the graph of f. Answer: 44 LR == 4.0 . The graph of f is a triangle of area 4. OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 21. Compute the left and right Riemann sums—L6 and R6, respectively—for f x x ( ) = − − ( ) 33 on   0, 6 . Compute their average value and compare it with the area under the graph of f. Answer: 66 LR == 9.000 . The graph of f is a triangle with area 9. 22. Compute the left and right Riemann sums—L4 and R4, respectively—for ( ) 2 f x x =−4 on   −2, 2 and compare their values. Answer: 44 LR == 5.4641 . The graph of f is a semicircle of area 2 6.28   . 23. Compute the left and right Riemann sums—L6 and R6, respectively—for ( ) ( )2 f x x = − − 93 on   0,6 and compare their values. Answer: 66 LR == 13.12899 . They are equal. Express the following endpoint sums in sigma notation but do not evaluate them. 24. L30 for f x( ) = x 2 on   1, 2 Answer: 2 30 30 1 11 1 30 30 i i L =  − =+    25. L10 for f x( ) = 4 - x 2 on   −2, 2 Answer: ( ) 2 10 10 1 4 1 4 2 4 10 10 i i L =  − = − − +    26. R20 for f x( ) = sin x on   0,  Answer: 20 20 1 sin 20 20 i i R   =  =    27. R100 for ln x on   1, e Answer: ( ) 100 100 1 1 ln 1 1 100 100 i ei Re = −  = + −    OpenStax Calculus Volume 1 Instructor Answer and Solution Guide In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Compare the left and right Riemann sums. Which one is larger? 28. [T] L100 and R100 for 2 y x x = − +3 on the interval   −1, 1 Answer: 100 L = 6.727 , 100 R = 6.607 , L100 R100. The plot shows that the left endpoint sum is an overestimate because the function is decreasing. Similarly, the right Riemann sum is an underestimate. Thus, the area lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles. 29. [T] L100 and R100 for 2 yx = on the interval   0,1 Answer: 100 R = 0.33835 , 100 L = 0.32835. The plot shows that the left Riemann sum is an underestimate because the function is increasing. Similarly, the right Riemann sum is an overestimate. The area lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles. OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 30. [T] L50 and R50 for 2 1 1 x y x + = − on the interval   2, 4 Answer: 50 L =1.1121, 50 R =1.0854 . The plot shows that the left Riemann sum overestimates the area because the function is decreasing. Similarly, the right endpoint sums are underestimates. The area lies between the left and right endpoint approximations. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles. 31. [T] L100 and R100 for 3 yx =+ 2 on the interval   −1, 1 Answer: 100 L = 3.980 , 100 R = 4.020 , R100 L100. The left endpoint sum is an underestimate because the function is increasing. Similarly, a right endpoint approximation is an overestimate. The area lies between the left and right endpoint estimates. OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 32. [T] L50 and R50 for yx = tan( ) on the interval 0, 4     Answer: 50 L = 0.3387, 50 R = 0.3544 . The plot shows that the left Riemann sum is an underestimate because the function is increasing. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles. 33. [T] L100 and R100 for 2x ye = on the interval   −1, 1 Answer: 100 L = 3.555 , 100 R = 3.670 . The plot shows that the left Riemann sum is an underestimate because the function is increasing. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles. 34. Let tj denote the time that it took Tejay van Garteren to ride the jth stage of the Tour de France in 2014. If there were a total of 21 stages, interpret 21 1 j j t =  . Answer: The sum represents the total time it took van Garteren to complete the 2014 Tour de France. 35. Let j r denote the total rainfall in Portland on the jth day of the year in 2009. Interpret 31 1 j j r =  . Answer: The sum represents the cumulative rainfall in January 2009. OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 36. Let j d denote the hours of daylight and j  denote the increase in the hours of daylight from day j −1 to day j in Fargo, North Dakota, on the jth day of the year. Interpret 365 1 2 j j d  = + . Answer: The sum represents the total number of hours of daylight on the 365th day of the year in Fargo. 37. To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds 1 10 mi to his daily routine each week, what is the total mileage on Joe’s shoes after 25 weeks? Answer: The total mileage is ( ) 25 1 1 71 i 10 i =  − +    7 7 25 12 25 10 =  +   = 385 mi . 38. The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide (CO2) each decade since 1960, in parts per million (ppm). Estimate the total increase in atmospheric CO2 between 1964 and 2013. Average Annual Atmospheric CO2 Increase, 1964–2013 Decade Ppm/y 1964–1973 1.07 1974–1983 1.34 1984–1993 1.40 1994–2003 1.87 2004–2013 2.07 Answer: Add the rates and multiply by 10 to get 77.5 ppm. 39. The following table gives the approximate increase in sea level in inches over 20 years starting in the given year. Estimate the net change in mean sea level from 1870 to 2010. Approximate 20-Year Sea Level Increases, 1870–1990 Starting Year 20-Year Change 1870 0.3 1890 1.5 1910 0.2 1930 2.8 1950 0.7 1970 1.1 1990 1.5 Answer: Add the numbers to get 8.1-in. net increase. OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 40. The following table gives the approximate increase in dollars in the average price of a gallon of gas per decade since 1950. If the average price of a gallon of gas in 2010 was $2.60, what was the average price of a gallon of gas in 1950? Approximate 10-Year Gas Price Increases, 1950–2000 Starting Year 10-Year Change 1950 0.03 1960 0.05 1970 0.86 1980 –0.03 1990 0.29 2000 1.12 Answer: Subtract the increase of $2.32 from $2.60 to get 28¢ in 1950. 41. The following table gives the percent growth of the U.S. population beginning in July of the year indicated. If the U.S. population was 281,421,906 in July 2000, estimate the U.S. population in July 2010. Annual Percentage Growth of U.S. Population, 2000–2009 Year % Change/Year 2000 1.12 2001 0.99 2002 0.93 2003 0.86 2004 0.93 2005 0.93 2006 0.97 2007 0.96 2008 0.95 2009 0.88 (Hint: To obtain the population in July 2001, multiply the population in July 2000 by 1.0112 to get 284,573,831.) Answer: 309,389,957 In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8. 42. Answer: 8 L = + + + + + + + = 1 2 3 4 5 4 3 2 2 This OpenStax Calculus Volume 1 – IASG Chapters 5–6 Instructor Answer and Solution Guide is designed to support understanding of key calculus concepts related to derivatives and early integration topics. It includes structured solutions covering applications of derivatives, such as optimization and related rates, as well as foundational ideas in integration and the Fundamental Theorem of Calculus. This guide is ideal for students preparing for university calculus exams who need clear, step-by-step explanations to strengthen problem-solving skills in mid-level calculus topics.