Work

Work, in physics, is simply the energy it takes to push or lift something some distance. All we're really saying is: push harder, or push farther, and it takes more work. With a constant force that's just force times distance. The formula handles a force that changes: $F(x)$ (the force at position $x$), $dx$ (a tiny step, short enough that the force barely changes) and $\int_a^b$ (add up force times step over the whole trip). Hooke's law, $F = kx$, is just the force a spring pushes back with: $k$ (how stiff the spring is) times $x$ (how far you've stretched it).

This is the same move we've made all unit. Area was height times width, until the height changed; then we sliced it into thin rectangles and integrated. Work is force times distance, until the force changes; then we slice the motion into short steps and integrate. On a graph of force against position, the work is literally the area under the curve.

Units: newton-meters (joules) or foot-pounds.

Springs. Stretching a spring takes more force the farther it's stretched, $F = kx$. Find $k$ from one measurement, then integrate.

Pumping liquids. Here the force doesn't change, but the distance does. Slice the liquid into thin horizontal layers. Each layer has a weight (density × $g$ × volume) and must be lifted a different distance: the layers near the bottom travel farther. Work for a layer = weight × distance lifted; integrate over all the layers.

Cables and chains. Each piece of a hanging chain is lifted a different distance; slice the chain the same way.

Engineers size pumps for water towers, calculate the energy stored in springs and bows, and estimate the work to lift elevator cables, all with $W = \int F\,dx$.