For the following exercises (1-6), find the work done.

**1. **Find the work done when a constant force [latex]F=12[/latex] lb moves a chair from [latex]x=0.9[/latex] to [latex]x=1.1[/latex] ft.

**2.** How much work is done when a person lifts a 50 lb box of comics onto a truck that is 3 ft off the ground?

**3. **What is the work done lifting a 20 kg child from the floor to a height of 2 m? (Note that 1 kg equates to 9.8 N)

**4.** Find the work done when you push a box along the floor 2 m, when you apply a constant force of [latex]F=100\text{N}.[/latex]

**5. **Compute the work done for a force [latex]F=\frac{12}{{x}^{2}}[/latex] N from [latex]x=1[/latex] to [latex]x=2[/latex] m.

**6. **What is the work done moving a particle from [latex]x=0[/latex] to [latex]x=1[/latex] m if the force acting on it is [latex]F=3{x}^{2}[/latex] N?

For the following exercises (7-11), find the mass of the one-dimensional object.

**7. **A wire that is 2 ft long (starting at [latex]x=0)[/latex] and has a density function of [latex]\rho (x)={x}^{2}+2x[/latex] lb/ft

**8. **A car antenna that is 3 ft long (starting at [latex]x=0)[/latex] and has a density function of [latex]\rho (x)=3x+2[/latex] lb/ft

**9. **A metal rod that is 8 in. long (starting at [latex]x=0)[/latex] and has a density function of [latex]\rho (x)={e}^{1\text{/}2x}[/latex] lb/in.

**10. **A pencil that is 4 in. long (starting at [latex]x=2)[/latex] and has a density function of [latex]\rho (x)=5\text{/}x[/latex] oz/in.

**11. **A ruler that is 12 in. long (starting at [latex]x=5)[/latex] and has a density function of [latex]\rho (x)=\text{ln}(x)+(1\text{/}2){x}^{2}[/latex] oz/in.

For the following exercises (12-16), find the mass of the two-dimensional object that is centered at the origin.

**12. **An oversized hockey puck of radius 2 in. with density function [latex]\rho (x)={x}^{3}-2x+5[/latex]

[latex]\frac{332\pi }{15}[/latex]

**13. **A frisbee of radius 6 in. with density function [latex]\rho (x)={e}^{\text{−}x}[/latex]

**14. **A plate of radius 10 in. with density function [latex]\rho (x)=1+ \cos (\pi x)[/latex]

**15.** A jar lid of radius 3 in. with density function [latex]\rho (x)=\text{ln}(x+1)[/latex]

**16. **A disk of radius 5 cm with density function [latex]\rho (x)=\sqrt{3x}[/latex]

**17. **A 12-in. spring is stretched to 15 in. by a force of 75 lb. What is the spring constant?

**18. **A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm?

**19. **A 1-m spring requires 10 J to stretch the spring to 1.1 m. How much work would it take to stretch the spring from 1 m to 1.2 m?

**20. **A spring requires 5 J to stretch the spring from 8 cm to 12 cm, and an additional 4 J to stretch the spring from 12 cm to 14 cm. What is the natural length of the spring?

**21. **A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?

**22. **A force of [latex]F=20x-{x}^{3}[/latex] N stretches a nonlinear spring by [latex]x[/latex] meters. What work is required to stretch the spring from [latex]x=0[/latex] to [latex]x=2[/latex] m?

**23. **Find the work done by winding up a hanging cable of length 100 ft and weight-density 5 lb/ft.

**24. **For the cable in the preceding exercise, how much work is done to lift the cable 50 ft?

**25. **For the cable in the preceding exercise, how much additional work is done by hanging a 200 lb weight at the end of the cable?

**26. [T]** A pyramid of height 500 ft has a square base 800 ft by 800 ft. Find the area [latex]A[/latex] at height [latex]h.[/latex] If the rock used to build the pyramid weighs approximately [latex]w=100{\text{lb/ft}}^{3},[/latex] how much work did it take to lift all the rock?

**27. [T]** For the pyramid in the preceding exercise, assume there were 1000 workers each working 10 hours a day, 5 days a week, 50 weeks a year. If the workers, on average, lifted 10 100 lb rocks 2 ft/hr, how long did it take to build the pyramid?

**28. [T]** The force of gravity on a mass [latex]m[/latex] is [latex]F=\text{−}((GMm)\text{/}{x}^{2})[/latex] newtons. For a rocket of mass [latex]m=1000\text{kg},[/latex] compute the work to lift the rocket from [latex]x=6400[/latex] to [latex]x=6500[/latex] km. (*Note*: [latex]G=6×{10}^{-17}{\text{N m}}^{2}\text{/}{\text{kg}}^{2}[/latex] and [latex]M=6×{10}^{24}\text{kg}\text{.})[/latex]

**30. [T]** For the rocket in the preceding exercise, find the work to lift the rocket from [latex]x=6400[/latex] to [latex]x=\infty .[/latex]

**31. [T]** A rectangular dam is 40 ft high and 60 ft wide. Compute the total force [latex]F[/latex] on the dam when

- the surface of the water is at the top of the dam and
- the surface of the water is halfway down the dam.

**32. [T]** Find the work required to pump all the water out of a cylinder that has a circular base of radius 5 ft and height 200 ft. Use the fact that the density of water is 62 lb/ft^{3}.

**33. [T]** Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

**34. [T]** How much work is required to pump out a swimming pool if the area of the base is 800 ft^{2}, the water is 4 ft deep, and the top is 1 ft above the water level? Assume that the density of water is 62 lb/ft^{3}.

**35. **A cylinder of depth [latex]H[/latex] and cross-sectional area [latex]A[/latex] stands full of water at density [latex]\rho .[/latex] Compute the work to pump all the water to the top.

**36. **For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.

**37. **A cone-shaped tank has a cross-sectional area that increases with its depth: [latex]A=(\pi {r}^{2}{h}^{2})\text{/}{H}^{3}.[/latex] Show that the work to empty it is half the work for a cylinder with the same height and base.