Cycle/closed integration question

[Jamessamuel]

Hello,
im sorry the picture is upside down. But my problem is with this formula:

The book i am reading from says the net work done in moving a piston which must push gases, varying the pressure and volume is given by the formula shown in the picture. He then reduces it to what he calls a "cycle integral".I have a few questions/problems:
1.Why, if there are 2 distinct curves present is he using the same equation/ integral for both?
2. surely, if the limits are arranged like so, the whole thing should reduce to zero?
this is exactly how the book showed it.

Regards,

James.

 

[HallsofIvy]

I'm not sure what notation is used but those two integrals should be over the two different paths. If the integrand, p (pressure?), is an "exact derivative" (in physics a "conservative" function), meaning that there exist some function, F(x,y), such that $\nabla F= p$ then each integral would just be F(p;2)- F(p1) and F(p1)- F(p2) so there sum is 0. But if the integrand is NOT exact the integral along two different paths will not be the same.