I have the following problem:
Calculate the line integral of $$\int_C xds$$ where $C$ is the parabola $y=x^2$ with $x \in [-1,1]$
I know that for a line integral we need the norm of the derivative of the parametric function of the parabola. The problem is that I don't know if I parametrized correctly the parabola since the integral results in $0$. The parametrization I made was $r(t) = (\frac{1}{2}t, \frac{1}{4}t^2)$.
Let $x=t$ and $y=t^2$ then it is a injective parametrizatión of $C$, note that $\frac{dx}{dt}=1$ and $\frac{dy}{dt}=2t$ now note that $t\in [-1,1]$, and now you can transfor the integral of line in that new integral like a $\int_{-1}^{1} t dt=\frac{1}{2}t^2|_{-1}^{1}$ and now you can evaluate it.