Help for understaning the formula for integration a Line integral of first kind

Question

I need help to understand this formula: $$\int_Гf(x,y,z)dl= \int_{t1}^{t2}f(\varphi(t), \psi(t), \mathcal X(t)) \sqrt{(\varphi^\prime)^2t \ + \ (\psi^\prime)^2t \ + \ \mathcal (X^\prime)^2}dt$$ First from reading textbook and some stuff on the net i didn't understand what is that transition from dl to dt in theory, one of the two must be the length of the line that is integrated i think and the other question is what exactly is the parameter $\mathcal X$. For example if have a integral of first kind given like follows : $$\int_C(x^2+y^2-2z)dl$$ given by the curved line :$$C:x=4cos2t, y=4sin2t\ , z=6t;\ t\in[0,3\pi].$$ Can i use that formula because this integral is from first kind and from my understanding so far its parametrized already or i am wrong. And second what i should put for the parameter $\mathcal X.$ Thanks for any help in advance.

Answer 1

Yes, you're right that this is parameterized already. The functions $\varphi(t), \psi(t), \text{and } \mathcal X(t)$ are the parameterizations of $x,y,z$, which are the inputs to our function $f(x,y,z)$.
The integral $$\int_C(x^2+y^2-2z)dl$$ has integrand $f(x,y,z)=x^2+y^2-2z$. Use $$\varphi(t) = x(t) = 4\cos{2t}, \psi(t) = y(t) = 4\sin{2t}, \mathcal X(t) = z(t) = 6t$$ so that it fits into the form of your first equation. $\mathcal X(t)$ is just the third input $z$ to the function $f(x,y,z)$.