Evaluate ? c(y2+2x+1)dx+(2xy+4y? 1)dy along the given path c from (0,0) to (3,9). c is the parabolic path y=x2

Let . Then




which means represents a conservative vector field. This means there is some scalar function such that , and so the gradient theorem applies. This means the value of the line integral is path independent. We have









and so we have



Then the value of the line integral is