Work

Work is energy transferred to or from an object via the application of force by a displacement, measured in joules ($J$). Work is interlinked with energy.

$$U={\int }_{s_1}^{s_2}\vec{F}\cdot \overrightarrow{ds}$$ $$U=Fs\cos \theta$$

For a spring, where the direction of displacement is opposite to the restorative spring force:

$$U_{1\cdot 2}=\frac{1}{2}k(x_1^2-x_2^2)$$

For a particle moving some displacement:

$$U_{1\cdot 2}=-mg(y_2-y_1)$$

Integrals

Riemann sums are super useful to build definite integrals for work. Often times we deal with pumping some fluid out of a volume.

$$W=Fd=mgh=V\rho gh=\int Vg\rho hdy$$

When discussing circulation line integrals, the work done by a vector field to move an object on a curve is given by the integral.