Work is force times distance, and when the force changes along the way it becomes an integral. For a spring, Hooke's law makes the force kx; for a rope or a tank, each thin slice moves its own distance.
A spring has natural length 21 cm. A force of 110 N stretches it to 22 cm. How much work does it take to stretch it from 31 cm to 43 cm?
A force of F(x) = 3x2+4x+10 pounds moves an object along a line from x = 1 ft to x = 4 ft. Find the work done.
A 70 ft rope weighing 0.5 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?
A spring has natural length 15 cm. A force of 40 N stretches it to 18 cm. How much work does it take to stretch it from 20 cm to 22 cm?
A force of F(x) = x2+2x pounds moves an object along a line from x = 1 ft to x = 5 ft. Find the work done.
An 80 ft rope weighing 3 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?
A spring has natural length 12 cm. A force of 135 N stretches it to 15 cm. How much work does it take to stretch it from 18 cm to 28 cm?
A force of F(x) = 2x2+7x+10 pounds moves an object along a line from x = 0 ft to x = 3 ft. Find the work done.
A 100 ft rope weighing 0.5 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?
A spring has natural length 13 cm. A force of 55 N stretches it to 18 cm. How much work does it take to stretch it from 14 cm to 23 cm?
A force of F(x) = x2+x+6 pounds moves an object along a line from x = 2 ft to x = 3 ft. Find the work done.
A cylindrical tank with radius 2 ft and height 12 ft is full of water (62.4 lb per cubic foot). How much work does it take to pump all the water out over the top? Leave the answer in terms of π
A spring has natural length 23 cm. A force of 65 N stretches it to 28 cm. How much work does it take to stretch it from 30 cm to 42 cm?
A force of F(x) = 3x2+8x+10 pounds moves an object along a line from x = 3 ft to x = 6 ft. Find the work done.
A 20 ft rope weighing 4 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?
A spring has natural length 12 cm. A force of 105 N stretches it to 21 cm. How much work does it take to stretch it from 22 cm to 31 cm?
A force of F(x) = 3x2+2x+10 pounds moves an object along a line from x = 0 ft to x = 3 ft. Find the work done.
A 60 ft rope weighing 0.5 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?
A spring has natural length 15 cm. A force of 135 N stretches it to 19 cm. How much work does it take to stretch it from 20 cm to 27 cm?
A force of F(x) = 3x2+3x+1 pounds moves an object along a line from x = 0 ft to x = 4 ft. Find the work done.
A spring has natural length 21 cm. A force of 110 N stretches it to 22 cm. How much work does it take to stretch it from 31 cm to 43 cm?211.2 J
A force of F(x) = 3x2+4x+10 pounds moves an object along a line from x = 1 ft to x = 4 ft. Find the work done.123 ft⋅lb
A 70 ft rope weighing 0.5 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?1225 ft⋅lb
A spring has natural length 15 cm. A force of 40 N stretches it to 18 cm. How much work does it take to stretch it from 20 cm to 22 cm?1.6 J
A force of F(x) = x2+2x pounds moves an object along a line from x = 1 ft to x = 5 ft. Find the work done.3196 ft⋅lb
An 80 ft rope weighing 3 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?9600 ft⋅lb
A spring has natural length 12 cm. A force of 135 N stretches it to 15 cm. How much work does it take to stretch it from 18 cm to 28 cm?49.5 J
A force of F(x) = 2x2+7x+10 pounds moves an object along a line from x = 0 ft to x = 3 ft. Find the work done.2159 ft⋅lb
A 100 ft rope weighing 0.5 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?2500 ft⋅lb
A spring has natural length 13 cm. A force of 55 N stretches it to 18 cm. How much work does it take to stretch it from 14 cm to 23 cm?5.445 J
A force of F(x) = x2+x+6 pounds moves an object along a line from x = 2 ft to x = 3 ft. Find the work done.689 ft⋅lb
A cylindrical tank with radius 2 ft and height 12 ft is full of water (62.4 lb per cubic foot). How much work does it take to pump all the water out over the top? Leave the answer in terms of π17971.2π ft⋅lb
A spring has natural length 23 cm. A force of 65 N stretches it to 28 cm. How much work does it take to stretch it from 30 cm to 42 cm?20.28 J
A force of F(x) = 3x2+8x+10 pounds moves an object along a line from x = 3 ft to x = 6 ft. Find the work done.327 ft⋅lb
A 20 ft rope weighing 4 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?800 ft⋅lb
A spring has natural length 12 cm. A force of 105 N stretches it to 21 cm. How much work does it take to stretch it from 22 cm to 31 cm?15.225 J
A force of F(x) = 3x2+2x+10 pounds moves an object along a line from x = 0 ft to x = 3 ft. Find the work done.66 ft⋅lb
A 60 ft rope weighing 0.5 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?900 ft⋅lb
A spring has natural length 15 cm. A force of 135 N stretches it to 19 cm. How much work does it take to stretch it from 20 cm to 27 cm?20.08125 J
A force of F(x) = 3x2+3x+1 pounds moves an object along a line from x = 0 ft to x = 4 ft. Find the work done.92 ft⋅lb
A spring has natural length 10 cm, and 20 N stretches it to 15 cm. How much work stretches it from its natural length to 20 cm?
k = 20 N divided by 0.05 m, which is 400 N/m.
Stretching 10 cm means x runs from 0 to 0.1 m.
The integral of 400x from 0 to 0.1 is 200 times 0.01.
The answer is 2 J.
Where students go wrong
Measuring the stretch in centimeters. The spring constant is in newtons per meter, so convert first, and measure from the natural length, not from zero on the ruler.
Also called work integrals, Hooke's law problems or pumping problems.
What you can put on this worksheet
Springs
A spring has natural length 18 cm. A force of 150 N stretches it to 24 cm. How much work does it take to stretch it from its natural length to 20 cm?→ 0.5 J
A variable force
A force of F(x) = 7x pounds moves an object along a line from x = 2 ft to x = 5 ft. Find the work done.→ 2147 ft⋅lb
Ropes and pumping
A 70 ft rope weighing 2 lb per foot hangs from the top of a building. How much work does it take to pull the whole rope up to the top?→ 4900 ft⋅lb
Questions about these worksheets
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Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
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That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the work problems with integrals questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
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You pick from 3: springs, a variable force and ropes and pumping. Tick as many as you want and set how many of each, or let it spread them evenly.
Measuring the stretch in centimeters. The spring constant is in newtons per meter, so convert first, and measure from the natural length, not from zero on the ruler.