Consider the parametrized curve r(u)=(5+u, 4u+7, 3u). Which one of the following functions is the arc length parametrization of the curve which starts from s=0 at the point (4,3,-3) along the curve?
I have calculated r'(u)= (1,4,3) and thus ||r'(u)||=√ 26.
However I'm confused on what to do next to find s(u). I understand there should be some sort of integration next wrt u and that would give [√ 26*u].
Any help would be greatly appreciated!
The arc starts from $r(0)$ and ends in $r(-1)=(4, 3, -3)$. Hence the arc signed-length is the value of the integral $$\int_0^{-1} \lvert r'(u)\rvert\,du=\int_0^{-1} \sqrt{26}\,du=-\sqrt{26}$$ Similarly when computing the area of regions bounded by graphs of functions, the definite integral takes signed values. Anyway, the arc length is $\sqrt{26}$