Work is the energy transferred to an objection as an accumulation of force applied to that object over some displacement. E.g. lifting a box off the ground loads the box with potential energy in the form of force applied to counter gravity’s influence on the box. The units of work, the same as the SI units for energy, are Joules. If a constant \(\mathrm{force}\) is being applied over some \(\mathrm{distance}\) then the total work done, the total energy added, is the product \(\mathrm{force}\!\times\!\mathrm{distance}\) But if that force is not constant, we must use an integral to measure the total work.
Force is defined as the product of the mass of an object and the acceleration applied to that object: \(\bm{F} = m\bm{\alpha}.\) Suppose that a force \(\bm{F}\) is applied to some object, displacing it from position \(x=a\) to position \(x=b,\) and that the magnitude of that force in the direction of displacement is a function \(f\) of the position \(x\) between \(a\) and \(b\) — perhaps because the mass \(m\) is changing or because the acceleration \(\bm{\alpha}\) changing — so \(\bigl|\bm{F}\bigr| = f(x)\,.\) The total amount of work \(W\) done, the sum total of force accumulated as energy, is computed by the integral \[ W = \int\limits_a^b f(x)\,\mathrm{d}x \,.\]
Caution: The integral is taken with respect to displacement, not time; work does not care how long it takes to you displace an object.
A common example: Hooke’s Law can be used to calculate the amount of energy (work) loaded into an object attached to a spring as the spring is extended or compressed. Hooke’s Law states that the force required to hold a spring displaced from its natural position is proportional to the distance it’s displaced from that natural position. I.e. \(f(x) = kx\) where \(k\) is some constant depending on the spring.