In the example
Evaluate the line integral
\begin{equation}\int_C (x^2-2xy)dx + (y^2-2xy)dy \end{equation} where C is the parabola $y=x^2$ and -1 ≤ x ≤ 1
I paramterized the parabola to equal $r(t)= < t, t^2 >$ and then integrated with the following set up:
\begin{equation}\int_{-1}^1 (t^2-2t^3)dt + \int_{-1}^1 (t^4-2t^3)2t dt\end{equation}
when doing this I got an answer of $-14/15$.
I am not sure if it is correct because can a line integral be a negative value? If not, can someone please point out where I went wrong?
Thank you.
As per my calculations, $-\frac{14}{15}$ is the correct answers.
For the second part of your question, line integrals can take negative values. Intuitively speaking, line integrals represent the (amount of) work done by a vector field.
So, your vector field in this case is doing work in the opposite direction to your line integral. You can also think of your solution as a tangent pointing in the opposite direction to you line integral.
I'd also suggest taking a look at Chapter 1 (Vector Analysis) of Introduction to Electrodynamics by Griffiths. I think it provides a great intuition on Line, Surface, and Volume integrals.
Hope this helped!