Lets start by considering work done by the system. We know that work = force x distance, so if we consider a gas expanding against a constant pressure then work = pressure x volume pushed out, or 

w = - p.ΔV

The '-' sign arrives from the idea that if the final volume is larger than the first, (ΔV greater than zero) the gas has done work. Our definition of w is the work done on the system. Hence the '-' sign ensures consistency.

What if the external pressure was zero? This is called free expansion. Since 
w = - p.ΔV, and p=0, therefore w=0 also.

The work done is always 0 for expansion into a vacuum.

However, suppose the external pressure is not constant, then we must integrate the pressure with respect to the volume change, ie:

w = - ∫_Vi^Vf p dV∫_Vi^Vf p dV

Finally, what if the expansion took place in a diathermic container? The temperature would have remained the same, and if we also assume a perfect gas, we can use the equation of state to write pV = nRT. So p = nRT/V and our integral becomes

w = - ∫_Vi^Vf nRT/V dV

since nRT is a constant, we have

w = - nRT ∫_Vi^Vf V^-1 dV

Boyles Law gives us this pressure - volume graph. The work done is the area under the line. (a) w = - p.ΔV (b) w = - ∫ViVf p dV