Work problems with integrals worksheet

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.

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Sigma Prep · sigmaprep.io/worksheets Name Work Problems Date Period Solve each work problem. Solve spring work problems 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 spring has natural length 7 cm. A force of 140 N stretches it to 9 cm. How much work does it take to stretch it from its natural length to 14 cm? A spring has natural length 7 cm. A force of 125 N stretches it to 11 cm. How much work does it take to stretch it from 12 cm to 18 cm? A spring has natural length 28 cm. A force of 80 N stretches it to 30 cm. How much work does it take to stretch it from its natural length to 36 cm? A spring has natural length 6 cm. A force of 30 N stretches it to 12 cm. How much work does it take to stretch it from 11 cm to 18 cm? A spring has natural length 23 cm. A force of 45 N stretches it to 33 cm. How much work does it take to stretch it from its natural length to 29 cm? A spring has natural length 8 cm. A force of 165 N stretches it to 12 cm. How much work does it take to stretch it from 14 cm to 20 cm? Find work done by a variable force A force of $F (x) =$ $3 x^{2} + 4 x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 4$ ft. Find the work done. A force of $F (x) =$ $x^{2} + x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 5$ ft. Find the work done. A force of $F (x) =$ $x^{2} + 4 x + 7$ pounds moves an object along a line from $x = 3$ ft to $x = 5$ ft. Find the work done. A force of $F (x) =$ $x^{2} + 7 x$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done. A force of $F (x) =$ $2 x^{2} + x + 8$ pounds moves an object along a line from $x = 0$ ft to $x = 3$ ft. Find the work done. A force of $F (x) =$ $3 x^{2} + 8 x$ pounds moves an object along a line from $x = 1$ ft to $x = 3$ ft. Find the work done. A force of $F (x) =$ $x^{2} + x + 2$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done. Find the work to lift or pump 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 cylindrical tank with radius 1 ft and height 11 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 $π$ . An 80 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? A 50 ft rope weighing 1.5 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 cylindrical tank with radius 1 ft and height 10 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 $π$ . An 80 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? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Work Problems

Date Period

Solve each work problem.

Solve spring work problems
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 spring has natural length 7 cm. A force of 140 N stretches it to 9 cm. How much work does it take to stretch it from its natural length to 14 cm?
A spring has natural length 7 cm. A force of 125 N stretches it to 11 cm. How much work does it take to stretch it from 12 cm to 18 cm?
A spring has natural length 28 cm. A force of 80 N stretches it to 30 cm. How much work does it take to stretch it from its natural length to 36 cm?
A spring has natural length 6 cm. A force of 30 N stretches it to 12 cm. How much work does it take to stretch it from 11 cm to 18 cm?
A spring has natural length 23 cm. A force of 45 N stretches it to 33 cm. How much work does it take to stretch it from its natural length to 29 cm?
A spring has natural length 8 cm. A force of 165 N stretches it to 12 cm. How much work does it take to stretch it from 14 cm to 20 cm?

Find work done by a variable force
A force of $F (x) =$ $3 x^{2} + 4 x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 4$ ft. Find the work done.
A force of $F (x) =$ $x^{2} + x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 5$ ft. Find the work done.
A force of $F (x) =$ $x^{2} + 4 x + 7$ pounds moves an object along a line from $x = 3$ ft to $x = 5$ ft. Find the work done.
A force of $F (x) =$ $x^{2} + 7 x$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done.
A force of $F (x) =$ $2 x^{2} + x + 8$ pounds moves an object along a line from $x = 0$ ft to $x = 3$ ft. Find the work done.
A force of $F (x) =$ $3 x^{2} + 8 x$ pounds moves an object along a line from $x = 1$ ft to $x = 3$ ft. Find the work done.
A force of $F (x) =$ $x^{2} + x + 2$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done.

Find the work to lift or pump
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 cylindrical tank with radius 1 ft and height 11 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 $π$ .
An 80 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?
A 50 ft rope weighing 1.5 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 cylindrical tank with radius 1 ft and height 10 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 $π$ .
An 80 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?



Sigma Prep · sigmaprep.io/worksheets Name Work ProblemsAnswer keyVersion 1 Date Period Solve each work problem. Solve spring work problems 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 spring has natural length 7 cm. A force of 140 N stretches it to 9 cm. How much work does it take to stretch it from its natural length to 14 cm? $17.15$ J A spring has natural length 7 cm. A force of 125 N stretches it to 11 cm. How much work does it take to stretch it from 12 cm to 18 cm? $15$ J A spring has natural length 28 cm. A force of 80 N stretches it to 30 cm. How much work does it take to stretch it from its natural length to 36 cm? $12.8$ J A spring has natural length 6 cm. A force of 30 N stretches it to 12 cm. How much work does it take to stretch it from 11 cm to 18 cm? $2.975$ J A spring has natural length 23 cm. A force of 45 N stretches it to 33 cm. How much work does it take to stretch it from its natural length to 29 cm? $0.81$ J A spring has natural length 8 cm. A force of 165 N stretches it to 12 cm. How much work does it take to stretch it from 14 cm to 20 cm? $22.275$ J Find work done by a variable force A force of $F (x) =$ $3 x^{2} + 4 x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 4$ ft. Find the work done. $123$ ft $\cdot$ lb A force of $F (x) =$ $x^{2} + x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 5$ ft. Find the work done. $\frac{280}{3}$ ft $\cdot$ lb A force of $F (x) =$ $x^{2} + 4 x + 7$ pounds moves an object along a line from $x = 3$ ft to $x = 5$ ft. Find the work done. $\frac{236}{3}$ ft $\cdot$ lb A force of $F (x) =$ $x^{2} + 7 x$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done. $\frac{232}{3}$ ft $\cdot$ lb A force of $F (x) =$ $2 x^{2} + x + 8$ pounds moves an object along a line from $x = 0$ ft to $x = 3$ ft. Find the work done. $\frac{93}{2}$ ft $\cdot$ lb A force of $F (x) =$ $3 x^{2} + 8 x$ pounds moves an object along a line from $x = 1$ ft to $x = 3$ ft. Find the work done. $58$ ft $\cdot$ lb A force of $F (x) =$ $x^{2} + x + 2$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done. $\frac{112}{3}$ ft $\cdot$ lb Find the work to lift or pump 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 $\cdot$ lb A cylindrical tank with radius 1 ft and height 11 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 $π$ . $3775.2 π$ ft $\cdot$ lb An 80 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? $6400$ ft $\cdot$ lb A 50 ft rope weighing 1.5 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? $1875$ ft $\cdot$ lb A cylindrical tank with radius 1 ft and height 10 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 $π$ . $3120 π$ ft $\cdot$ lb An 80 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? $1600$ ft $\cdot$ lb Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Work ProblemsAnswer keyVersion 1

Date Period

Solve each work problem.

Solve spring work problems
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 spring has natural length 7 cm. A force of 140 N stretches it to 9 cm. How much work does it take to stretch it from its natural length to 14 cm? $17.15$ J
A spring has natural length 7 cm. A force of 125 N stretches it to 11 cm. How much work does it take to stretch it from 12 cm to 18 cm? $15$ J
A spring has natural length 28 cm. A force of 80 N stretches it to 30 cm. How much work does it take to stretch it from its natural length to 36 cm? $12.8$ J
A spring has natural length 6 cm. A force of 30 N stretches it to 12 cm. How much work does it take to stretch it from 11 cm to 18 cm? $2.975$ J
A spring has natural length 23 cm. A force of 45 N stretches it to 33 cm. How much work does it take to stretch it from its natural length to 29 cm? $0.81$ J
A spring has natural length 8 cm. A force of 165 N stretches it to 12 cm. How much work does it take to stretch it from 14 cm to 20 cm? $22.275$ J

Find work done by a variable force
A force of $F (x) =$ $3 x^{2} + 4 x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 4$ ft. Find the work done. $123$ ft $\cdot$ lb
A force of $F (x) =$ $x^{2} + x + 10$ pounds moves an object along a line from $x = 1$ ft to $x = 5$ ft. Find the work done. $\frac{280}{3}$ ft $\cdot$ lb
A force of $F (x) =$ $x^{2} + 4 x + 7$ pounds moves an object along a line from $x = 3$ ft to $x = 5$ ft. Find the work done. $\frac{236}{3}$ ft $\cdot$ lb
A force of $F (x) =$ $x^{2} + 7 x$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done. $\frac{232}{3}$ ft $\cdot$ lb
A force of $F (x) =$ $2 x^{2} + x + 8$ pounds moves an object along a line from $x = 0$ ft to $x = 3$ ft. Find the work done. $\frac{93}{2}$ ft $\cdot$ lb
A force of $F (x) =$ $3 x^{2} + 8 x$ pounds moves an object along a line from $x = 1$ ft to $x = 3$ ft. Find the work done. $58$ ft $\cdot$ lb
A force of $F (x) =$ $x^{2} + x + 2$ pounds moves an object along a line from $x = 0$ ft to $x = 4$ ft. Find the work done. $\frac{112}{3}$ ft $\cdot$ lb

Find the work to lift or pump
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 $\cdot$ lb
A cylindrical tank with radius 1 ft and height 11 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 $π$ . $3775.2 π$ ft $\cdot$ lb
An 80 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? $6400$ ft $\cdot$ lb
A 50 ft rope weighing 1.5 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? $1875$ ft $\cdot$ lb
A cylindrical tank with radius 1 ft and height 10 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 $π$ . $3120 π$ ft $\cdot$ lb
An 80 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? $1600$ ft $\cdot$ lb

How to do these

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 $F (x) =$ $7 x$ pounds moves an object along a line from $x = 2$ ft to $x = 5$ ft. Find the work done. → $\frac{147}{2}$ ft $\cdot$ 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 $\cdot$ lb

Questions about these worksheets

Yes. Every work problems with integrals sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

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.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

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.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

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.

Work problems with integrals worksheet

Select your difficulty

Solve spring work problems

Find work done by a variable force

Find the work to lift or pump

Solve spring work problems

Find work done by a variable force

Find the work to lift or pump

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these work problems with integrals worksheets free?+

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Can I get a different version of the same work problems with integrals worksheet?+

Is this a free alternative to Kuta Software for work problems with integrals?+

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