Cycle/closed integration question

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The discussion centers on the application of a formula for calculating the net work done in moving a piston under varying pressure and volume conditions. The user, James, questions the use of the same integral for two distinct curves and the implications of the limits potentially reducing the integral to zero. Another participant clarifies that if the pressure function is an exact derivative, the integrals over the two paths will yield a net result of zero. However, if the integrand is not exact, the results will differ based on the paths taken.

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

20151010_122144.jpg


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.
 
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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 [itex]\nabla F= p[/itex] 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.
 

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