Work by Variable Force

Definition — Work done by variable force $W = \int \vec{F} \cdot d\vec{r}$.
1D Case — $W = \int_{x_1}^{x_2} F(x) dx$.
Graphical — Area under $F-x$ curve (signed area).
Spring Force — $F_s = -kx$. Work done *by* external agent to stretch/compress from $x_1$ to $x_2$: $W = \frac{1}{2}k(x_2^2 - x_1^2)$.
Work-Energy Theorem — $W_{net} = \Delta K = K_f - K_i$.
Conservative Force — Work done is path-independent, $W = -Delta U$.
Units — Joules (J).

To calculate Work by a Variable Force, remember 'W.V.F. = Integrate F.dr'.

Work is Variable, so Force is Integrated, From Displacement Range.

Work = Integral
Force = $F(x)$ (or $\vec{F}$)
Displacement = $dx$ (or $d\vec{r}$)
Range = Limits of integration ($x_1$ to $x_2$)