Let $f(x,y)=2x-y$, and consider the trajectory $x=t^4$,$y=t^4$ with $-1\leq t\leq 1$.
a) Calculate the integral of $f$ along this trajectory.
b) Evaluate the function arc length $s(t)$ and do part a) in terms of $s$.
a) Let $\phi:[-1,1]\to \mathbb{R}^2$ be a trajectory defined as $\phi(t)=(t^4,t^4)$. If $z=f(x,y)$, then \begin{equation} \int_\phi zds=\int_{-1}^1 f(\phi(t))\|\phi'(t)\|dt, \end{equation} where $\|\phi'(t)\|=\sqrt{32t^6}=4\sqrt{2}|t|^3$ and $f(\phi(t))=t^4$. Then \begin{equation} \int_\phi zds=\int_{-1}^1 t^4\left(4\sqrt{2}|t|^3\right)dt=-4\sqrt{2}\int_{-1}^0 t^7dt+4\sqrt{2}\int_0^1t^7dt=\sqrt{2}. \end{equation}
Did I do well?
b) I have to parametrize a curve with respect to Arc Length, but the trajectory isn't regular, because $\phi'(0)=(0,0)$. So i don't know what to do.
Could you give me a hint? Please. Thank you.