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CALCULUS WORKSHEET ON RIEMANN SUMS AND ANTIDERIVATIVES Work the following on notebook paper. Use your calculator, and give decimal answers correct to three decimal places. Estimate the area bounded by the curve and the x-axis on the given interval using the indicated number of subintervals by finding: (a) a left Riemann sum (b) a right Riemann sum (c) a midpoint Riemann sum 1. y x [0,1] 2. y n = 4 subintervals 1 x [1, 3] n = 4 subintervals _______________________________________________________________________________________ 3. Estimate the area bounded by the curve and the x-axis on [1, 6] using the 5 equal subintervals by finding: (a) a left Riemann sum (b) a right Riemann sum (c) a midpoint Riemann sum _______________________________________________________________________________________ 4. Oil is leaking out of a tank. The rate of flow is measured every two hours for a 12-hour period, and the data is listed in the table below. Time (hr) Rate (gal/hr) 0 40 2 38 4 36 6 30 8 26 10 18 12 8 (a) Draw a possible graph for the data given in the table. (b) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a left Riemann sum with three equal subintervals. (c) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a right Riemann sum with three equal subintervals. (d) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period by finding a midpoint Riemann sum with three equal subintervals. _________________________________________________________________________________________ Evaluate. 5. x 1 2 x 3 dx 8. 3 x x 2 dx 11. x 6. x2 2 x 3 x4 dx 9. x 12. 3 sin 5x cos 5x dx 7. 5 sec x tan x sec x dx 2 10. 5 2 3 x 4 2 dx x4 x2 8x 1 2 cos 2 x3 dx dx _________________________________________________________________________________________ 13. Find a function f such that the graph of f has a horizontal tangent at (2, 0) and f x 6 x . AP CALCULUS WORKSHEET ON DEFINITE INTEGRALS AND AREA Work the following on notebook paper. Do not use your calculator except on problem 15. Evaluate. 1. 2. 0 x 3. 4. 1 0 5. x3 1 dx 6. 3 x x2 1 dx 1 2 1 2 4 1 dx 3 7. 2 x 1 dx 8. 2 2 0 9 x 6 9. 0 10. 6 2 dx cos 3x dx 11. sin 2x dx 12. 12 6 2 x 0 0 25 x 2 dx 4 x x 5 0 0 2 2x cos dx 3 sin 3 x cos x dx 4 tan x sec 2 x dx 6 sec 2 x tan 2 x dx ________________________________________________________________________________________ 13. Find the area bounded by the graph of f x 2sin x sin 2 x and the x-axis on the interval 0, . x and the x-axis 2 14. Find the area bounded by the graph of f x sec 2 2 on the interval , . 2 3 ________________________________________________________________________________________ Use your calculator on problem 15. 15. The rate at which water is being pumped into a tank is given by the function R t . A table of selected values of R t , for the time interval 0 t 20 minutes, is shown below. t (min.) R t (gal/min) 0 25 4 28 9 33 17 42 20 46 (a) Use data from the table and four subintervals to find a left Riemann sum to approximate the value of 0 R t dt . 20 (b) Use data from the table and four subintervals to find a right Riemann sum to approximate the value of 0 R t dt . 20 (c) A model for the rate at which water is being pumped into the tank is given by the function W t 25e 0.03t , where t is measured in minutes and W t is measured in gallons per minute. Use the model to find the value of 20 0 W t dt . CALCULUS WORKSHEET ON ALGEBRAIC & U-SUBSTITUTION Work the following on notebook paper. Do not use your calculator. Evaluate. 2x 1. x x 2 dx 5. 2. x 6. x 3 x 1 dx 3. 2 x 1 x dx 4. x 4 2 dx 9. 7 x 2 2 dx 1 x dx x4 7. 8. 1 1 1 3 x x 2 1 dx 2 1 2 x 0 11. 4 x3 1 dx 1 dx 2x 1 4 10. 2 3 3 x dx x5 x dx 2x 1 5 ________________________________________________________________________________________ 12. Find the area bounded by the graph of y x 3 x 1 and the x-axis on the interval [0, 7]. 13. Find the area bounded by the graph of y x cos x and the x-axis on the interval , . 3 2 ________________________________________________________________________________________ 14. Solve: dy 48 given that 1, 3 is a point on the solution curve. 3 dx 3x 5 ________________________________________________________________________________________ Given that f x is an even function and that 15. 2 f x dx 0 16. 8 0 f x dx 3 , find: 2 2 f x dx 2 17. 2 3 f x dx 0 ________________________________________________________________________________________ Given that f x is an odd function and that 18. 2 f x dx 0 19. 8 0 f x dx 3 , find: 2 2 f x dx 2 20. 2 3 f x dx 0 ________________________________________________________________________________________ 1 1 3 3 2 5n ... n n n n n 21. Write lim 3 as a definite integral, given that n is a positive integer. 22. The closed interval [c, d] is partitioned into n equal subintervals, each of width x, by the numbers n xk n c x0 , x1, ..., xn where x0 x1 x2 ... xn 1 d . Write lim i 1 2 x as a definite integral. CALCULUS WORKSHEET 1 ON FUNDAMENTAL THEOREM OF CALCULUS Work the following on notebook paper. Work problems 1 - 2 by both methods. Do not use your calculator. 1 1. y 2 2 and y 1 6. Find y 3 . x . 4 2. f x cos 2 x and f 0 3. Find f ________________________________________________________________________________________ Work problems 3 – 7 using the Fundamental Theorem of Calculus and your calculator. 3. f x cos x3 and f 0 2. Find f 1 . 4. f x e x and f 5 1. Find f 2 . 2 5. A particle moving along the x-axis has position x t at time t with the velocity of the particle v t 5sin t 2 . At time t = 6, the particle’s position is (4, 0). Find the position of the particle when t = 7. 6. Let F t represent a bacteria population which is 4 million at time t = 0. After t hours, the population is growing at an instantaneous rate of 2t million bacteria per hour. Find the total increase in the bacteria population during the first three hours, and find the population at t = 3 hours. 7. A particle moves along a line so that at any time t 0 its velocity is given by v t t . At time 1 t2 t = 0, the position of the particle is s 0 5. Determine the position of the particle at t = 3. ______________________________________________________________________________________ Use the Fundamental Theorem of Calculus and the given graph. 8. The graph of f is shown on the right. y 1 f x dx 6.2 4 and f 1 3. Find f 4 . x 1 9. The graph of f is the semicircle shown on the right. 4 y Find f 4 given that f 4 7. x -4 10. The graph of f , consisting of two line segments and a semicircle, is shown on the right. Given that f 2 5 , find: (a) f 1 (b) f 4 (c) f 8 4 y x TURN->>> 11. Region A has an area of 1.5, and (a) 2 f x dx (b) 0 f x dx 0 f x dx 3.5. 6 Find: 6 6 12. The graph on the right shows the rate of change of the quantity of water in a water tower, in liters per day, during the month of April. If the tower has 12,000 liters of water in it on April 1, estimate the quantity of water in the tower on April 30. 13. A cup of coffee at 90° C is put into a 20° C room when t = 0. The coffee’s temperature is changing at a rate of r t 7e 0.3t C per minute, with t in minutes. Estimate the coffee’s temperature when t = 10. 14. Use the figure on the right and the fact that F 2 3 to sketch the graph of F x . Label the values of at least four points. CALCULUS WORKSHEET 2 ON FUNDAMENTAL THEOREM OF CALCULUS Work these on notebook paper. Use your calculator on problems 3, 8, and 13. 1. If f 1 12, f is continuous, and 2. If 1 f x dx 17, what is the value of f 4 ? 4 2 2 f x 3 dx 17, find 2 f x dx. 5 5 3. Water is pumped out of a holding tank at a rate of 5 5e 0.12t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later? 4. Given the values of the derivative f x in the table and that f 0 100, estimate f x for x = 2, 4, 6. Use a right Riemann sum. x f x 0 10 2 18 4 23 6 25 5. Consider the function f that is continuous on the interval [ 5,5] and for which 0 f x dx 4. Evaluate: 5 (a) f x 2 dx 0 5 (b) 3 2 f x 2 dx (c) 5 f x dx (f is even) (d) 5 f x dx (f is odd) 5 5 6. Use the figure on the right and the fact that P 0 2 to find values of P when t = 1, 2, 3, 4, and 5. 7. Using the figure on the right, sketch a graph of an antiderivatives G t of g t satisfying G 0 5. Label each critical point of G t with its coordinates. TURN->>> 8. Find the value of F 1 , where F x e x and F 0 2. 2 2 x, x 1 9. Given f x . Evaluate: 2, x 1 12 f x dx. 5 10. A bowl of soup is placed on the kitchen counter to cool. The temperature of the soup is given in the table below. Time t (minutes) Temperature T x (°F) (a) Find 0 105 5 99 8 97 12 93 0 T x dx . 12 (b) Find the average rate of change of T x over the time interval t = 5 to t = 8 minutes. 11. The graph of f which consists of a line segment and a semicircle, is shown on the right. Given that f 1 4, find: y (a) f 2 x (b) f 5 12. (Multiple Choice) If f and g are continuous functions such that g x f x for all x , f x dx 3 then 2 (A) g 2 g 3 (B) g 3 g 2 (D) f 3 f 2 (E) f 3 f 2 13. (Multiple Choice) If the function (C) g 3 g 2 f x is defined by f x x3 2 and g is an antiderivatives of f such that g 3 5 , then g 1 (A) 3.268 (B) 1.585 (C) 1.732 (D) 6.585 14. (Multiple Choice) The graph of f is shown in the figure at right. f x dx 2.3 and F x f x , then F 3 F 0 3 If 1 (A) 0.3 (B) 1.3 (C) 3.3 (D) 4.3 (E) 5.3 (E) 11.585 CALCULUS WORKSHEET ON AVERAGE VALUE Work the following on notebook paper. Use your calculator on problems 3 – 6, and give decimal answers correct to three decimal places. On problems 1 and 2, (a) Find the average value of f on the given interval. (b) Find the value of c such that f AVE f c . 1. f x x 3 , 2, 5 2. f x 2 x , 0, 4 __________________________________________________________________________________________ 3. The table below gives values of a continuous function. Use a midpoint Riemann sum with three equal subintervals to estimate the average value of f on [20, 50]. x f x 20 42 25 38 30 31 35 29 40 35 45 48 50 60 __________________________________________________________________________________________ 4. The velocity graph of an accelerating car is shown on the right. (a) Estimate the average velocity of the car during the first 12 seconds by using a midpoint Riemann sum with three equal subintervals. (b) At what time was the instantaneous velocity equal to the average velocity? __________________________________________________________________________________________ 5. In a certain city, the temperature, in °F, t hours after 9 AM was modeled by the function t T t 50 14sin . Find the average temperature during the period from 9 AM to 9 PM. 12 __________________________________________________________________________________________ 6. If a cup of coffee has temperature 95°C in a room where the temperature is 20°C, then, according to Newton’s Law of Cooling, the temperature of the coffee after t minutes is given by the function T t 20 75e t 50 . What is the average temperature of the coffee during the first half hour? __________________________________________________________________________________________ 7. Suppose the C t represents the daily cost of heating your house, measured in dollars per day, where t is time measured in days and t = 0 corresponds to January 1, 2010.. Interpret 90 90 1 0 C t dt and 90 0 0 C t dt . TURN->>> y 8. Using the figure on the right, (a) Find 1 f x dx . 6 (b) What is the average value of f on [1, 6]? x Graph of f ________________________________________________________________________________________ 9. The average value of y f x equals 4 for 1 x 6 and equals 5 for 6 x 8 . What is the average value of f x for 1 x 8 ? ________________________________________________________________________________________ 10. Suppose 0 f x dx 6 . 3 (a) What is the average value of f x on the interval x = 0 to x = 3? (b) If f x is even, what is the value of 3 f x dx ? 3 What is the average value of f x on the interval x 3 to x = 3? (c) If f x is odd, what is the value of 3 f x dx ? 3 What is the average value of f x on the interval x 3 to x = 3? ________________________________________________________________________________________ In problems 11 – 14, find the average value of the function on the given interval without integrating. Hint: Use Geometry. (No calculator) x 4, 4 x 1 on 4, 2 x 2, 1 x 2 y 11. f x x 12. f x 1 1 x 2 1, 1 y x 13. f x sin x, 0, 2 14. f x tan x, , 4 4