Stewart - Calculus 8e ET Chapter 6. All Answers

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1. Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midp ... oint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds. 2. Find the volume common to two spheres, each with radius r = if the center of each sphere lies on the surface of the other sphere. 3. Find the volume of the solid obtained by rotating about the x-axis the region under the curve from x = to x = . 4. Find the volume of the solid obtained by rotating the region bounded by and about the y-axis. 5. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. 6. Find the work done in pushing a car a distance of m while exerting a constant force of N.Stewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 7. The tank shown is full of water. Given that water weighs 62.5 lb/ft and R = 5, find the work (in lb-ft) required to pump the water out of the tank. 8. Find the average value of the function on the given interval. 9. The velocity v of blood that flows in a blood vessel with radius and length l at a distance from the central axis is , where P is the pressure difference between the ends of the vessel and q is the viscosity of the blood. Find the average velocity (with respect to r) over the interval 10. Find the number(s) a such that the average value of the function on the interval is equal to . 11. Sketch the region bounded by the graphs of the given equations and find the area of that region. , , , 12. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = + 4, y = 2x + 1 13. Find the area of the region bounded by the given curves (a) using integration with respect to x and (b) using integration with respect to y. y = 2x3, y = 5x + 6, x = 0Stewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 14. Use a graphing utility to (a) plot the graphs of the given functions and (b) find the approximate x-coordinates of the points of intersection of the graphs. Then find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the x-axis. Round answers to two decimal places. y = x5, y = 3x2 – x3 15. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. , , , ; the x-axis 16. Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. , y = 0, x = 1, x = 12; the y-axis 17. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = , y = x – 1; the line x = 2 18. Find the volume of the solid obtained by revolving the region under the graph of on [1, ) about the x-axis. 19. Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the y-axis. Round answers to two decimal places. y = 6x, y = – , x  0Stewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20. Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = 4 , y = 8x – 4, y = 16; the y-axisStewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. 1144 15 ft 2. 3. 4. 5. 6. ft-lb 7. 981,748 ft-lb 8. 9. 10. 11. 1 2 x 10 –10 y e  53Stewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 12. –5 x –20 y 13. a. 14 b. 14 14. a. 1 x 4 3 2 1 –1 y b. (0, 0) and (1.81, 3.90), 20.15 15. 16.Stewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2 4 6 8 10 12 x y 17. 1 5 1 2 3 4 5 6 x 1 y 18. 460 21Stewart - Calculus 8e ET Chapter 6 Form A © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 19. a. 1 2 3 x 9 8 7 6 5 4 3 2 1 10 y b. (0, 0) and (1.66, 9.98) c. 38.23 20. 9 1 2 3 x 5 10 15 20 y Stewart - Calculus 8e ET Chapter 6 Form B © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1. Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds. 2. Use the Midpoint Rule with n = to approximate the area of the region bounded by the given curves. 3. Find the volume common to two spheres, each with radius r = if the center of each sphere lies on the surface of the other sphere. 4. Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.Stewart - Calculus 8e ET Chapter 6 Form B © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 5. The base of S is the parabolic region Cross-sections perpendicular to the y axis are squares. Find the volume of S. 6. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. 7. Find the volume of the solid obtained by rotating the region bounded by and about the y-axis. 8. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. 9. Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. 10. Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x axis. 11. Find the work done in pushing a car a distance of m while exerting a constant force of N. 12. Find the average value of the function on the given interval. 13. The velocity v of blood that flows in a blood vessel with radius and length l at a distance from the central axis is , where P is the pressure difference between the ends of the vessel and q is the viscosity of the blood. Find the average velocity (with respect to r) over the intervalStewart - Calculus 8e ET Chapter 6 Form B © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 14. Find the average value of the function on the interval . 15. Find the number(s) a such that the average value of the function on the interval is equal to . 16. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = + 4, y = 2x + 1 17. Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = , y = 0, x = 3; the y-axis 18. Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the y-axis. Round answers to two decimal places. y = 6x, y = – , x  0 19. A chain weighing 4 lb/ft hangs vertically from a winch located 11 ft above the ground, and the free end of the chain is just touching the ground. Find the work done by the winch in pulling in the whole chain. 20. A tank has the shape of an inverted right circular cone with a base radius of 4 m and a height of 9 m. If the tank is filled to a height of 1 m, find the work required (to the nearest joule) to empty the tank by pumping the water over the top of the tank. (The mass of water is 1000 kg/m3 and the force of gravity is 9.8 m/sec2.)Stewart - Calculus 8e ET Chapter 6 Form B © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. 1144 15 ft 2. 3. 4. 5. 6. 7. 8. 9. 10. 512 11. ft-lb 12. 13. 14. 15. 16. –5 x –20 yStewart - Calculus 8e ET Chapter 6 Form B © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17. 81 2  1 2 3 x 8 6 4 2 10 12 14 16 18 y 18. a. 1 2 3 x 9 8 7 6 5 4 3 2 1 10 y b. (0, 0) and (1.66, 9.98) c. 38.23 19. 242 ft-lb 20. 16,724 JStewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Select the correct answer for each question. ____ 1. Graph the region between the curves and use your calculator to compute the area correct to five decimal places. a. 12.08127 b. 91.504 c. 3.141257 d. 3.66016 e. 18.3008 ____ 2. Find the area of the region bounded by the given curves. a. b. c. d. e. None of theseStewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 3. Sketch the region bounded by the graphs of the given equations and find the area of their region. , , a. –1 1 x 1 –1 y c. –1 1 x 1 –1 y b. –1 1 x 1 –1 y d. –1 1 x 1 –1 yStewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 4. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 6, y = 8, x = – , x = a. –1 1 x 8 6 4 2 y  – 2 c. –1 1 x 8 6 4 2 y – 1 b. –1 1 x 8 6 4 2 y 8 d. –1 1 x 8 6 4 2 y –Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 5. Use integration to find the area of the triangle with the given vertices. (0, 0), (1, 7), (4, 2) a. 13 2 b. 26 c. 28 d. 13 ____ 6. Sketch the region enclosed by . Find the area of the region. a. 32 3 b. c. d. 32 3 e.Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 7. Find the volume of the solid that is obtained by revolving the region about the line y = 5 2 . y =5/2 y = x y = x3 1 x 3 2 1 y a. 89 42 b. 89 14 c. 89 84 d. 89 ____ 8. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. – = 16, x  0, y = – 4, y = 4 a. 256 b. 512 3 c. 128 d. 256 3Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 9. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = cos x + 1, x = 0, y = 0, x = 1 2   a. 3 4  b. 3 4  c.  d.  ____ 10. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 – , y = 0; the line y = 5 a. 368 15 b. 184 5 c. 184 15 d. 92 15Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 11. Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the specified axis. a. b. c. d. e.Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 12. Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. , , , the x-axis a. 135 c. 45 b. 90 d. 180Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 13. Use cylindrical shells to find the volume of the solid. A sphere of radius . a. b. c. d. e.Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 14. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height t as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of . a. b. c. d. e. ____ 15. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = 2sin x + 1, x = 0, y = 0, x =   a.  b.  c. 3 d. 3Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 16. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 – x2, y = 0; the line y = 3 a. 104 15  b. 208 15  c. 104 5  d. 52 15 ____ 17. A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from cm to cm? a. b. c. d. e. ____ 18. A tank is full of water. Find the work required to pump the water out of the outlet. a. b. c. d. e. None of theseStewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 19. A spring has a natural length of cm. If a force of N is required to keep it stretched to a length of cm, how much work is required to stretch it from cm to cm? a. b. c. d. e. ____ 20. Newton's Law of Gravitation states that two bodies with masses attract each other with a force where r is the distance between the bodies and G is the gravitation constant. If one of bodies is fixed, find the work needed to move the other from to . a. b. c. d. e.Stewart - Calculus 8e ET Chapter 6 Form C © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. E 2. B 3. A 4. A 5. D 6. A 7. C 8. B 9. A 10. C 11. E 12. B 13. B 14. C 15. D 16. A 17. D 18. B 19. E 20. BStewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Select the correct answer for each question. ____ 1. Find the area of the region bounded by the given curves. a. 2 b. 2 3 c. 4 d. e. ____ 2. Graph the region between the curves and use your calculator to compute the area correct to five decimal places. a. 18.3008 b. 3.66016 c. 91.504 d. 3.141257 e. 12.08127Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 3. Find the area of the shaded region. y = x + 4 y –x = 2 – 4 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x 7654321 –1 –2 –3 –4 –5 –6 –7 –8 –9 y a. 100 3 b. 25 3 c. 50 9 d. 50 3Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 4. Find the area of the shaded region. y = y = x – –7 x 2 + 7x /x _ ` 1 2 x 3 2 1 –1 y a. 2 3 b. 4 3 c. 1 7 d. 8Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 5. Sketch the region bounded by the graphs of the given equations and find the area of their region. , , a. –1 1 x 1 –1 y c. –1 1 x 1 –1 y b. –1 1 x 1 –1 y d. –1 1 x 1 –1 yStewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 6. Sketch the region bounded by the graphs of the given equations and find the area of their region. , , a. 10 x 10 –10 y 19 c. 10 x 10 –10 y  19 3 b. 10 x 10 –10 y 38 3 d. 10 x 10 –10 y  38 3Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 7. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 4, y = 6, x = – , x = a. –1 1 x 6 4 2 –2 y – c. –1 1 x 6 4 2 –2 y  – 2 b. –1 1 x 6 4 2 –2 y 6 d. –1 1 x 6 4 2 –2 y – 1Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 8. Find the volume of the solid obtained by rotating the region bounded by about the x-axis. a. b. c. d. e. ____ 9. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. a. b. c. d. e. NoneStewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 10. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. a. b. c. d. e. None of these ____ 11. The height of a monument is m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side meters. Find the volume of the monument. a. b. c. d. e.Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 12. Find the volume of the solid that is obtained by revolving the region about the x-axis. y = x – ( 2 ) 2 + 3 1 2 3 4 5 6 7 8 x 8 6 4 2 10 12 14 16 18 20 y a. 129 5 b. 258 5 c. 387 10 d. 129 10 ____ 13. Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves. Rounded to the nearest hundredth. a. b. c. d. e.Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 14. The base of a solid is a circular disk with radius . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base. a. 49 b. 16 c. 6 d. 7 e. None of these ____ 15. Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. The choices are rounded to the nearest hundredth. a. b. c. d. e. ____ 16. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x + 1, x = 0, y = 0, x =   a.  b. 3 2 c. 3 2 d. Stewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 17. A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from cm to cm? a. b. c. d. e. ____ 18. If J of work are needed to stretch a spring from cm to cm and another J are needed to stretch it from cm to cm, what is the natural length of the spring? Round the answer to nearest integer. a. b. c. d. e. ____ 19. An aquarium m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the facts that the density of water is ) a. b. c. d. e. ____ 20. A particle moves a distance of 160 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 10 lb in a direction opposite to that of the motion. What is the work done by the force? a. –1,600 ft-lb b.  1 1600 ft-lb c.  1 16 ft-lb d. 16 ft-lbStewart - Calculus 8e ET Chapter 6 Form D © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. B 2. A 3. D 4. B 5. B 6. B 7. C 8. B 9. B 10. C 11. D 12. B 13. E 14. E 15. E 16. C 17. D 18. B 19. E 20. AStewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 1. Graph the region between the curves and use your calculator to compute the area correct to five decimal places. Select the correct answer. a. 12.08127 b. 91.504 c. 3.141257 d. 3.66016 e. 18.3008 2. Find the area of the shaded region. y = x + 4 y –x = 2– 4 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x 654321 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 y 3. Sketch the region bounded by the graphs of the given equations and find the area of their region. , , 4. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 6, y = 8, x = – , x =Stewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 5. Sketch the region enclosed by . Find the area of the region. Select the correct answer. a. 32 3 b. c. d. 32 3 e. 6. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. 7. The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. about the y-axis 8. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 – , y = 0; the line y = 5 9. Find the volume of the solid obtained by rotating the region bounded by about the lineStewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 10. Find the volume of a pyramid with height and base an equilateral triangle with side a = . Select the correct answer. a. b. c. d. e. 11. Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. The choices are rounded to the nearest hundredth. 12. Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2Stewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 13. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height t as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of . ____ 14. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. Select the correct answer. x2 – y2 = 81, x  0, y = –9, y = 9 a. 972 b. 1458 c. 1944 d. 2916 15. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 – x2, y = 0; the line y = 3Stewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 16. Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line. 1 2 3 4 5 x 5 4 3 2 1 y y-axis ____ 17. A tank is full of water. Find the work required to pump the water out of the outlet. Select the correct answer. a. b. c. d. e. None of these 18. A spring has a natural length of cm. If a force of N is required to keep it stretched to a length of cm, how much work is required to stretch it from cm to cm? 19. If J of work are needed to stretch a spring from cm to cm and another J are needed to stretch it from cm to cm, what is the natural length of the spring? Round the answer to nearest integer.Stewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 20. Evaluate the integral. Select the correct answer. a. b. c. d.Stewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. E 2. 112 3 3. –1 1 x 1 –1 y 4. –1 1 x 8 6 4 2 y  – 2 5. A 6. 7. 8. 184 15Stewart - Calculus 8e ET Chapter 6 Form E © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 9. 10. A 11. 12. 1 x 1 y x-axis 13. 14. A 15. 104 15  16. 128 3  17. B 18. 19. 20. BStewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 1. Sketch the region enclosed by the curves Find the area of the region correct to two decimal places. Select the correct answer. a. b. c. d. e. f. 2. Find the area of the region bounded by the given curves. 3. Find the number b such that the line divides the region bounded by the curves and into two regions with equal area. 4. Find the area of the shaded region. y = x + 1 y –x = 2– 1 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x 654321 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 yStewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 5. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. Select the correct answer. a. b. c. d. e. None of these 6. The height of a monument is m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side meters. Find the volume of the monument. ____ 7. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. Select the correct answer. – = 1, x  0, y = – 1, y = 1 a. 8 3 b. 4 c. 4 3 d. 2 8. Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves. Rounded to the nearest hundredth. 9. The base of a solid is a circular disk with radius . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.Stewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 10. Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. The choices are rounded to the nearest hundredth. Select the correct answer. a. b. c. d. e. 11. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 9 – x2, y = 0; the line ____ 12. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Select the correct answer. y = 1 – , y = 0; the line y = 6 a. 112 15 b. 224 5 c. 448 15 d. 224 15Stewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 13. Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the specified axis. Select the correct answer. a. b. c. d. e. 14. Use cylindrical shells to find the volume of the solid. A sphere of radius .Stewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 15. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height p as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of . Select the correct answer. a. b. c. d. e. 16. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 4 – x2, y = 0; the line y = 6 17. A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from cm to cm? 18. A heavy rope, ft long, weighs lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?Stewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 19. A spring has a natural length of cm. If a force of N is required to keep it stretched to a length of cm, how much work is required to stretch it from cm to cm? Select the correct answer. a. b. c. d. e. 20. A steel girder weighing 800 lb is hoisted from ground level to the roof of a 30-ft building using a chain that weighs 2 lb/running foot. Find the work done.Stewart - Calculus 8e ET Chapter 6 Form F © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. B 2. 2 3 3. 4. 40 3 5. A 6. 7. A 8. 9. None of these 10. E 11. 216 –8 –6 –4 –2 2 4 6 8 x 10 20 30 –10 y 12. D 13. B 14. 15. E 16. 1408 15  17. 18. 19. C 20. 24,900 ft-lbStewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1. Find the area of the region bounded by the given curves. ____ 2. Find the number b such that the line divides the region bounded by the curves and into two regions with equal area. Select the correct answer. a. b. c. d. e. 3. Find the area of the shaded region. y = y = x – –6 x 2 + 6x /x _ ` 1 2 x 3 2 1 –1 y 4. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 6, y = 8, x = – , x =Stewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 5. Sketch the region enclosed by . Find the area of the region. Select the correct answer. a. 32 3 b. c. d. 32 3 e. 6. Find the volume of the solid obtained by rotating the region bounded by about the x-axis. 7. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. ____ 8. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. Select the correct answer. a. b. c. d. e. None of theseStewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 9. Find the volume of the solid that is obtained by revolving the region about the line y = 5 2 . y =5/2 y = x y = x3 1 x 3 2 1 y 10. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. – = 16, x  0, y = – 4, y = 4 11. Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves. Rounded to the nearest hundredth. 12. Find the volume of a pyramid with height and base an equilateral triangle with side a = .Stewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 13. The base of a solid is a circular disk with radius . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base. Select the correct answer. a. 6 b. 7 c. 16 d. 49 e. None of these 14. Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the specified axis. 15. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 16 – , y = 0; the line x = – 4 16. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height t as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of .Stewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 17. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. Select the correct answer. y = 2sin x + 1, x = 0, y = 0, x =   a.  b.  c. 3 d. 3 18. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 – x2, y = 0; the line y = 3 ____ 19. A bucket weighs lb and a rope of negligible weight are used to draw water from a well that is ft deep. The bucket starts with 50 lb of water and is pulled up at a rate of 10 ft/s, but water leaks out of a hole in the bucket at a rate of 0.5 lb/s. Find the work done in pulling the bucket to the top of the well. Select the correct answer. a. b. c. d. e. 20. A particle moves a distance of 190 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work done by the force?Stewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. 2. D 3. 7 6 4. –1 1 x 8 6 4 2 y  – 2 5. A 6. 7. 8. D 9. 89 84 10. 512 3 11. 12. 13. E 14. 15. 2,048 3 Stewart - Calculus 8e ET Chapter 6 Form G © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. –8 –6 –4 –2 2 4 6 8 x 10 20 30 y 16. 17. D 18. 104 15  19. A 20. –2,850 ft-lbStewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1. Sketch the region enclosed by the curves Find the area of the region correct to two decimal places. ____ 2. Graph the region between the curves and use your calculator to compute the area correct to five decimal places. Select the correct answer. a. 12.08127 b. 91.504 c. 3.141257 d. 3.66016 e. 18.3008 3. Find the area of the region bounded by the given curves. 4. Find the area of the shaded region. y = y = x – –6 x 2 + 6x /x _ ` 1 2 x 3 2 1 –1 y 5. Sketch the region bounded by the graphs of the given equations and find the area of their region. , , 6. Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 6, y = 8, x = – , x =Stewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 7. Use integration to find the area of the triangle with the given vertices. Select the correct answer. (0, 0), (1, 7), (4, 2) a. 13 2 b. 26 c. 28 d. 13 8. Sketch the region enclosed by . Find the area of the region. ____ 9. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Select the correct answer. a. b. c. d. e. NoneStewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ____ 10. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. Select the correct answer. a. b. c. d. e. None of these 11. The height of a monument is m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side meters. Find the volume of the monument. ____ 12. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. Select the correct answer. – = 16, x  0, y = – 4, y = 4 a. 256 b. 512 3 c. 128 d. 256 3 ____ 13. The base of a solid is a circular disk with radius . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base. Select the correct answer. a. 6 b. 7 c. 16 d. 49 e. None of theseStewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 14. Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = , y = –x + 4; the y-axis 15. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 16 – , y = 0; the line x = – 4 ____ 16. Use cylindrical shells to find the volume of the solid. A sphere of radius . Select the correct answer. a. b. c. d. e. ____ 17. A spring has a natural length of cm. If a force of N is required to keep it stretched to a length of cm, how much work is required to stretch it from cm to cm? Select the correct answer. a. b. c. d. e.Stewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 18. If J of work are needed to stretch a spring from cm to cm and another J are needed to stretch it from cm to cm, what is the natural length of the spring? Round the answer to nearest integer. ____ 19. An aquarium m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the facts that the density of water is ) Select the correct answer. a. b. c. d. e. 20. A particle moves a distance of 190 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work done by the force?Stewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Answer Key 1. 2. E 3. 4. 7 6 5. 10 x 10 –10 y 88 3 6. –1 1 x 8 6 4 2 y  – 2 7. D 8. 32 3 9. B 10. DStewart - Calculus 8e ET Chapter 6 Form H © 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 11. 12. B 13. E 14. 64 3  –5 –4 –3 –2 –1 1 2 3 4 5 x 5 4 3 2 1 –1 –2 –3 –4 –5 y 15. 2,048 3  –8 –6 –4 –2 2 4 6 8 x 10 20 30 y 16. B 17. E 18. 19. E 20. –2,850 ft-lb [Show More]

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