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The problem is
1) Show that the line integral is independent of the path.
2) Find the integral from (1, 0) to (0, 4)
A line integral such as that is independent of the path if and only if the differential being integrated is an "exact differential"- that is, that there exist a function F(x,y) such that . Since we must have [tex]F_{xy}= F_{yx}, that, in turn, is true if and only if . Is that true?
Once you have shown that, you can do (2) by actually finding that F(x,y).
We must have . Integrating that, treating y as a constant, . (Since we are treating y as a constant, the "constant of integration" may be a function of y- that is the "g(y)".)
From and so . The " " cancels out- that had to happen in order that g' be a function of of y only and happens precisely because of the condition in (1). That tells us g'(y)= -2y so that where C now really is a constant. That is, . Evaluate that at (0,4) and (1, 4) and subtract.