Work and Fluid Force

Key idea

Integrals compute work and hydrostatic force.

In depth

Work is force times distance; when force varies, W = ∫ F(x) dx. For pumping liquids, slice into layers, compute each layer’s weight times distance lifted. Fluid pressure = ρgh grows with depth.

Worked example

A spring with F = 5x stretched from 0 to 2 m: W = ∫₀² 5x dx = 10 J.

Formula

Real-world uses

• Calculating the energy needed to pump water out of a tank or the force on a dam wall

Common mistake

Using total distance for every layer instead of each layer’s own distance.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Related Rates – Link rates of change of connected quantities.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.