Q: A particle moves along the $$x-$$axis from $$x=1 to$$ $$x=3$$ under a force $$f(x) = x^2$ + 2x (in $$pounds). What is the total work done?

$$W = \frac{50}{3}\ \text{ft-lb}$$

Question 1

A 5-lb bucket is lifted from the ground to the top of a 20-ft building using a rope that weighs 0.08 lb/ft. How much work is done in lifting both the bucket and the rope to the top of the building?

Accepted Answer

$$W = 116 	ext{ ft-lb}$$

Question 2

A force of $$40\ 	ext{N} is $$required to stretch a spring from its natural length of $$10\ 	ext{cm} to$$ $$15\ 	ext{cm}. $$What is the spring constant $$k in$$ $$	ext{N/m}$$?

Accepted Answer

$$k = 800\ 	ext{N/m}$$

Question 3

How much work is done in stretching the spring in the previous question from $$15\ 	ext{cm} to$$ $$18\ 	ext{cm}$$?

Accepted Answer

$$W = 1.56\ 	ext{J}$$

Question 4

A particle moves along the $$x-$$axis from $$x=1 to$$ $$x=3$$ under a force $$f(x) = x^2$ + 2x (in $$pounds). What is the total work done?

Accepted Answer

$$W = \frac{50}{3}\ 	ext{ft-lb}$$

Question 5

Which of the following integrals correctly represents the work required to pump all the water to the top of an inverted conical tank of height $$5\ 	ext{m}$$ and base radius $$2\ 	ext{m}$$, filled to a depth of $$4\ 	ext{m}$$, with water density $$1000\ 	ext{kg/m}^3$$ and $$g=9.8\ 	ext{m/s}^2$$?

Accepted Answer

$$W = \$$int_0$$^4 9800 \cdot \frac{4\pi}{25} y^2$ (5-y) \, dy$$

Question 6

If a force $$F(x)$$ varies with position $$x$$, which formula gives the exact work done in moving an object from $$x=a to$$ $$x=b$$?

Accepted Answer

$$W = \int_a^b F(x)\,dx$$

Question 7

A $$1.2\ 	ext{kg}$$ book is lifted $$0.7\ 	ext{m}$$ vertically. What is the work done against gravity? ($$g = 9.8\ 	ext{m/s}^2)$$

Accepted Answer

$$W = 8.232\ 	ext{J}$$

Question 8

Which of the following best describes Hooke’s Law for springs?

Accepted Answer

$$f(x) = kx$$, where $$k is $$the spring constant and $$x is $$the displacement$

Question 9

A $$20\ 	ext{lb}$$ weight is lifted $$6\ 	ext{ft}$$ off the ground. How much work is done?

Accepted Answer

$$W = 120\ 	ext{ft-lb}$$

Question 10

In a lifting problem, a $$10\ 	ext{m}$$ cable weighing $$2\ 	ext{kg/m} is $$lifted to the top of a building. What is the integral to compute the work done (using $$g=9.8\ 	ext{m/s}^2$$)?

Accepted Answer

$$W = \$$int_0$$^{10} 2 \cdot 9.8 (10-x)\,dx$$

Question 11

Which of the following is NOT a typical application of definite integrals in work problems?

Accepted Answer

Calculating the area under a curve

Question 12

In the context of pumping problems, what does the expression $$(	ext{weight density}) 	imes (	ext{area}) 	imes (	ext{distance})$$ represent?

Accepted Answer

The infinitesimal work required to move a thin slice of liquid

Question 13

A spring has a natural length of $$0.2\ 	ext{m}$$ and a spring constant $$k=500\ 	ext{N/m}. $$How much work is required to stretch it to $$0.25\ 	ext{m}$$?

Accepted Answer

$$W = 3.125\ 	ext{J}$$

Question 14

Which of the following best explains why integration is needed for work problems involving lifting a rope?

Accepted Answer

Because the weight being lifted decreases as more rope is lifted

Question 15

A tank in the shape of an inverted cone has height $$5\ 	ext{m}$$ and base radius $$2\ 	ext{m}. If $$the tank is filled to a depth of $$4\ 	ext{m}$$, what is the radius of the water surface at that depth?

Accepted Answer

$$r = \frac{8}{5}\ 	ext{m}$$

Applications of Definite Integrals: Work, Springs, Lifting, and Pumping Problems

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