Work done by a variable force

Many real forces are not constant -- a spring pulls harder the more it stretches, and the force needed to compress a gas increases as the volume shrinks. When either the magnitude $F$ or the angle $\theta$ (or both) change as the body moves, the small work done for an infinitesimal displacement is still

but now $F\cos\theta$ is itself a function of the position, so the total work from $r_i$ to $r_f$ must be found by summing (integrating) all these small contributions:

$W = \int_{r_i}^{r_f} F\cos\theta\;dr$ …