how do you calculate work input?

Answer 1

Energy you put into the system is work input. It can be measured many ways.

Work done (W) to move an object is equal to force (F) provided to move that object a certain distance (s). So long as the force and path/distance are parallel
W=F s
Or if not parallel then
W=F s cos(angel between F and s)

Similarly we can show that the magnitude of the work done when a gas expands is therefore (or being forced to contract) equal to the product of the pressure (P) of the gas times the change in the volume (V2- V1) of the gas.
W=P(V2-V1) why you may ask.
Consider a cylinder of area A
As we already know W=F s and
F=P A (force equal pressure times area)
V=s A (Volume equal distance times area)
So
W= F s = P A (s2-s1) and since As2= V2 and As1=V1
This is why W=P(V2-V1).
The W=F s is the fundamental definition of mechanical work.