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6.4 Arc Length

6.4 Arc Length. Length of a Curve in the Plane. If y=f(x) s a continuous first derivative on [a,b], the length of the curve from a to b is. Example:. 6.5 Area of a Surface. Generated by revolving y=f(x). about the x-axis. Generated by revolving y=f(x). about the y-axis. Example:.

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6.4 Arc Length

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  1. 6.4 Arc Length

  2. Length of a Curve in the Plane If y=f(x) s a continuous first derivative on [a,b], the length of the curve from a to b is

  3. Example:

  4. 6.5 Area of a Surface

  5. Generated by revolving y=f(x) about the x-axis

  6. Generated by revolving y=f(x) about the y-axis

  7. Example:

  8. 6.6 Work Generally Work=force distance

  9. Example A force of 112 N is required to slide a cement block 4 m. How much work is necessary?

  10. “work” category 6.6 Lifting

  11. Lifting a bucket of sand • Sand weighs 144 lb. • Bucket weighs 4 lb. • Rope weighs 0.08 lb/ft. • Bucket leaks at a steady rate and the sand is ½ gone when the bucket is lifted 18 ft.

  12. How Much Work a) To lift the sand?

  13. How Much Work b) To lift the sand and the bucket?

  14. How Much Work c) To lift the sand, the bucket and the rope?

  15. As a bucket is raised 30 ft., water leaks out at a constant rate. Find the work done if the bucket originally contained 24 lb of water and 1/3 leaks out. Weight of empty bucket is 4 lb.

  16. How Much Work a) To lift the water?

  17. How Much Work To lift the water and bucket?

  18. Bucket • Lifted 20 ft • Contains 16 lb water ( 2 gal) • Leaks at constant rate • Empty when reaches top

  19. How Much Work • For water only? • For water and bucket, bucket weighs 5 lb • For water and bucket and rope, rope weighs 0.08 lb/ft.

  20. “work” category 6.6 Spring

  21. Hooke’s Lawthe force required to stretch a spring x units beyond is “natural” length is f(x)=k x, k is the constant of the spring.

  22. k= force length of stretch

  23. Example A force of 800 N stretches a spring 0.7 m. Find the Work done.

  24. Spring • 20 N force • Stretches spring 3 m beyond natural length How Much Work • To stretch from natural to 5 m? • To stretch one additional m?

  25. Spring • Natural length is 10 m • 8 N force stretches it 1.5 m How Much Work • To stretch from natural to 14 m? • To stretch from 11m to 13m?

  26. Spring 250N force stretches it 30 cm How Much Work to stretch from 20cm to 50cm?

  27. “work” category 6.6 Pumping

  28. Vertical cylindrical tank • Diameter = 3 ft • Height = 6ft • Contains water, 62.5 lb/ ft^3

  29. H M W to pump the water • Of a full tank out over the top of the tank?

  30. H M W to pump the water • Of a full tank thru a pipe which rises to a height of 4 ft above the top of the tank?

  31. H M W to pump the water • Of a ½ full tank over the top of the tank?

  32. H M W to pump the water • Of a ½ full tank out the pipe which rises to a height of 4 ft above the top of the tank?

  33. Cylindrical Tank • 4 m high • Radius = 2m • Tank buried so top is 1 m below graound level • Full of water, 1000 kg/m^3 • HMW to pump full tank to ground level?

  34. Cylindrical Tank • Radius = 5 ft • Height = 10 ft • Full of water, 62.5 lb/ft^3 • HMW to pump to a level 4 ft above ground?

  35. Above-ground circular pool • Diameter = 12 ft • Height = 5 ft • Depth of water is 4 ft • HMW to empty the pool by pumping the water over the top?

  36. A open tank has the shape of a right circular cone, full of water. Diameter of top = 8 ft, height = 6 ft HMW to empty by pumping over the top?

  37. A cylindrical gas tank has diameter 3 ft and is 4 ft long. The axis of the tank is horizontal.HMW to pump the entire contents into a tractor if the opening of the tractor tank is 5 ft above the top of the tank.

  38. “work” category 6.7 Fluid Force

  39. is the force of a fluid against a side of a tank.

  40. Example Find fluid force on the vertical side of the tank in the diagram, it’s full of water (62.5).

  41. Example • Glass tank, 3 ft. long • Square ends, 1 ft. long • Full of water (62.5) • Find force exerted on one end • Find force exerted on one side

  42. Example See diagram

  43. Right circular cylindrical tank • 90 ft high • 90 ft diameter • Full of molasses (100 lb/ft^3) • How much force is exerted on bottom 1 ft. band?

  44. Example • Sheet metal, area = 3 sq. ft. • Submerged horizontally in 5 ft. of water • Find the fluid force on the top side.

  45. Find the fluid force on the vertical side of a tank 3 ft. by 4 ft.It’s full of water (62.5)

  46. Find the fluid force on the vertical plate in the diagram..It’s submerged in water (1000 kg/cu.m.

  47. The vertical side of a form for poured concrete is 10 ft long and 2 ft high.Determine the force on this part of the form. Concrete weighs 140.7 lb/ft^3.

  48. Cylindrical gas tank, horizontal axis • Tank is ½ full • Diameter is 3 ft. • Gas weighs 42 lb/ft^3 • Find fluid force on an end of the tank.

  49. 6.8 Center of Mass

  50. 6.9 Distance traveled

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