When gas expands in a cylinder with radius 'r', the pressure at any given time is a function of the volume: P=P(V). The force exerted by the gas on the piston is the product of the pressure and the area: F=pi(r^2)P. Show that the work done by the gas when the volume expands from volume V1 to volume V2 is W= integral (V1 to V2) PdV. Medal to anyone who can help me show that this integral is correct and explain it!

Question

When gas expands in a cylinder with radius 'r', the pressure at any given time is a function of the volume: P=P(V).  The force exerted by the gas on the piston is the product of the pressure and the area: F=pi(r^2)P.  Show that the work done by the gas when the volume expands from volume V1 to volume V2 is W= integral (V1 to V2) PdV.  Medal to  anyone who can help me show that this integral is correct and explain it!