What is the price per gallon and how do we get the answer?
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We want to find a scalar function 
such that 
.
So we need to have
Integrating both sides with respect to
gives
Differentiating with respect to
gives
So we find that
By the fundamental theorem of calculus, we then know the line integral depends only on the values of at the endpoints of the path. Therefore
So we need to have
Integrating both sides with respect to
Differentiating with respect to
So we find that
By the fundamental theorem of calculus, we then know the line integral depends only on the values of