Find ∫c f*dr where f(x, y) = 〈 4xy3 , 6x2y2 〉 and c is a sequence of straight lines from (0,0) to (1,3) to (4,7) to (9,5) to (2,1

First note that is continuous, which means that there is some scalar function whose gradient is . In other words, is conservative, and so the gradient theorem applies here, which means the value of the line integral is given by .

So we look for such a function :












Thus by the gradient theorem, the value of the line integral is