12. An ideal spring has a natural length of 1 m. A force of 24 N...

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12. An ideal spring has a natural length of 1 m. A force of 24 N holds the spring stretched to a total length of 1.8 m. (a) Find the force constant k. (b) Determine the amount of work required to stretch the spring 2 m beyond its natural length. Answers: (a) k = 30 N/m. (b) W = 60 J. 13. Suppose a tank has the shape of a circular cylinder of radius 2 m and length 9 m and is filled with water. (a) Determine the amount of work required to pump the water to the ground level, assuming that the tank is oriented so that the circular cross sections are parallel to the ground and the top of the tank is 3 m below ground. (b) Determine the amount of work required to pump the water to the ground level, assuming that the tank is oriented so that the circular cross sections are perpendicular to the ground and the top of the tank is 3 m below ground. Answers: (a) W 8.32 106 J. (b) W 5.55 106 J. - 14. Let R be the region in the first quadrant below the curve y = 4 x2, and let be the solid obtained as R revolves about the y-axis. Assume that the density of R is 1 throughout. (a) Determine the area of R. (b) Determine the moment of R about the y-axis. (c) Determine the moment of R about the x-axis. (d) Determine the centroid of R. (e) Use Pappus's theorem to determine the volume of . Answers: (a) A = 18. (b) My 81/4 = 20.25. (c) Mx = 324/5 = 64.8. (d) (x,y) = (9/8, 18/5) = (1.125, 3.6) (e) V = 81 127.3 = 15. Let S be the line segment in the plane joining the points (2,1) and (10,7), and let be the surface obtained as S revolves about the x-axis. Determine the area of (a) using an integral and (b) using Pappus's theorem. Answer: A = 80 251.3. 16. Use an integral to verify that the volume of a sphere of radius 1 is 1/37. 17. Use an integral to verify that the surface area of a sphere of radius 1 is 47. 12. An ideal spring has a natural length of 1 m. A force of 24 N holds the spring stretched to a total length of 1.8 m. (a) Find the force constant k. (b) Determine the amount of work required to stretch the spring 2 m beyond its natural length. Answers: (a) k = 30 N/m. (b) W = 60 J. 13. Suppose a tank has the shape of a circular cylinder of radius 2 m and length 9 m and is filled with water. (a) Determine the amount of work required to pump the water to the ground level, assuming that the tank is oriented so that the circular cross sections are parallel to the ground and the top of the tank is 3 m below ground. (b) Determine the amount of work required to pump the water to the ground level, assuming that the tank is oriented so that the circular cross sections are perpendicular to the ground and the top of the tank is 3 m below ground. Answers: (a) W 8.32 106 J. (b) W 5.55 106 J. - 14. Let R be the region in the first quadrant below the curve y = 4 x2, and let be the solid obtained as R revolves about the y-axis. Assume that the density of R is 1 throughout. (a) Determine the area of R. (b) Determine the moment of R about the y-axis. (c) Determine the moment of R about the x-axis. (d) Determine the centroid of R. (e) Use Pappus's theorem to determine the volume of . Answers: (a) A = 18. (b) My 81/4 = 20.25. (c) Mx = 324/5 = 64.8. (d) (x,y) = (9/8, 18/5) = (1.125, 3.6) (e) V = 81 127.3 = 15. Let S be the line segment in the plane joining the points (2,1) and (10,7), and let be the surface obtained as S revolves about the x-axis. Determine the area of (a) using an integral and (b) using Pappus's theorem. Answer: A = 80 251.3. 16. Use an integral to verify that the volume of a sphere of radius 1 is 1/37. 17. Use an integral to verify that the surface area of a sphere of radius 1 is 47.