Work Done by a Variable Force

The simple formula $W=\vec{F}\cdot\vec{s}=Fs\cos\theta$ (with $\theta$ the angle between the applied force $\vec{F}$ and the displacement $\vec{s}$) is valid ONLY when both the force and the displacement are constant, finite vectors. In many real situations the force itself varies during the displacement -- gravitational force changes noticeably over very large (e.g. thousands-of-kilometres) displacements, and viscous/fluid-resistance forces depend on speed and so typically vary with time/position too. Direct application of $W=Fs\cos\theta$ is then invalid, and integration is required instead.

The method: split the total displacement (from $s_1$ to $s_2$) into a very large number of infinitesimally small displacements $ds$, each so small that the force F is practically constant over it (i.e. the change in F over that tiny interval cannot be meaningfully detected). The elementary work done over one such strip is $dW=F\,ds$; summing (integrating) all such elementary works over the whole path gives the total work, $W=\int_{s_1}^{s_2} F\,ds$ -- valid whenever the exact functional variation of F with s is known and is integrable. Geometrically, this integral equals the AREA under the force-versus-displacement (F-s) graph between $s_1$ and $s_2$, PROVIDED the force axis of the graph starts from zero; when the variation is linear, this area is simply that of a trapezium. …