Setting up work and fluid-force integrals
Slice the changing force into small contributions, express every dimension in one variable, and integrate with units attached.
How to recognize the method, run it, and know when it is the wrong choice.
Reviewed July 11, 2026Write one thin contribution as force times distance, or pressure times area. Then express density, depth, slice size, and travel distance in the same variable.
Work with a changing force
For constant force, work is force times distance. When force varies, partition the motion, approximate force on each small interval, and integrate the products.
Springs use Hooke’s law F = kx, while lifting chains or liquids also requires tracking how much material moves how far.
Fluid force varies with depth
Pressure equals weight density times depth. A horizontal strip at one depth has nearly constant pressure, so its force is pressure times strip area.
The depth is measured from the fluid surface, not automatically from the coordinate origin.
Draw and label one representative slice
The slice is the model. Label its width, thickness, depth, and movement distance before writing the integral. Most setup errors are visible immediately on that diagram.
A spring with k = 8 N/m is stretched from 0.1 m to 0.3 m beyond equilibrium. Find the work.
Use Hooke’s law with displacement from equilibrium.
Force changes throughout the motion.
Evaluate with meter-based bounds.
Common mistakes
- Using the spring’s total length instead of displacement from equilibrium.
- Treating fluid pressure as constant across changing depth.
- Mixing centimeters and meters inside one setup.
Three takeaways
- Build one differential contribution first.
- Depth and distance must be measured from the correct reference.
- Units are a powerful setup check.