How to calculate the line integral of a vector field over a parabola I am trying to answer a question about line integrals, I have had a go at it but I am not sure where I am supposed to incorporate the line integral into my solution.
$$ \mathbf{V} = xy\hat{\mathbf{x}} + -xy^2\hat{\mathbf{y}}$$
$$ \mathrm{d}\mathbf{l} = \hat{\mathbf{x}}\mathrm{d}x + \hat{\mathbf{y}}\mathrm{d}y $$
$$ \int_C\!\mathbf{V}\cdot\mathrm{d}\mathbf{l} = \int\!xy\,\mathrm{d}x - \int\!xy^2\,\mathrm{d}y = \left[ \frac{x^2y}{2}\right ]_?^? - \left[ \frac{xy^3}{3} \right]_?^? $$
I have a feeling that the parabola in question must come into play in the limits of the integrals, although I dont know how they are supposed to. The parabola in question is $y = \frac{x^2}{3}$ and the coordinates at which the line integral is supposed to go over are $a=(0,0)$ and $b=(3,3)$.

In this case, it's as simple as plugging in $y=\frac{x^2}{3}$ into the first integral and integrating from $x=0$ to $x=3$. In the second integral, do the same thing, but plug in $dy=\frac{2x}{3}dx$. In other words, parameterize the whole curve in terms of $x$.
In other examples (like circles), it may be more convenient to parameterize $x$ and $y$ with a different variable.
