Work as an Integral
Lesson notes
Overview
Work done by a variable force is the area under the force–distance curve. This topic is rated advanced and is number 189 of 245 in Mathematics. Deeper models, formal results and harder reasoning.
What you are seeing
The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
When a force is constant, work is force times distance. When the force changes with position, work is the area under the force–distance curve: W = ∫ F dx. Stretching a spring is the standard example, where F = kx gives W = ½kx². The same integral gives the work needed to lift a heavy cable or pump liquid from a tank, as each slice moves a different distance.
Reading the relationship
The expression "W = ∫ F dx" is a compact summary of the idea. It uses the symbols W, F, d, x; name what each one stands for before you start. Notice that it contains an integral, which accumulates a quantity over an interval. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.
Try this
Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Average Value of a Function”, the topic before it in Mathematics. Once you are comfortable, continue with “Centre of Mass of a Rod”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Work as an Integral” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Work as an Integral?
Work done by a variable force is the area under the force–distance curve. It is an advanced topic in Mathematics.
How do I use the Work as an Integral simulation?
The panel at the top shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression W = ∫ F dx packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Deeper models, formal results and harder reasoning. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Average Value of a Function”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.