Integration over curve

Question

I'm currently working on this problem :

Consider the curve $\Gamma \subset R^3$ with parametrization $\gamma$ such that : $$y(t):(r\cos t, r\sin t,ct)\quad \forall t \in [0,b ]$$ where $c>0$ and $b$ is a real number. Compute : $$\int _{\Gamma}x_1dx_1+x_2dx_2+x_3dx_3$$

I know the definition of the integral over a curve $\Gamma\subset R^n$ with parametrization $\gamma : [a,b] \rightarrow \Gamma$, with $\gamma\in C^1([a,b])$
$$\int_{\Gamma}fds=\int_{a}^{b}f(\gamma(t))\mid \dot \gamma(t)\mid dt$$ I don't know for this problem how to use this definition ? I mean, what is $f$ here ? Thanks for your help !

Answer 1

You are using the formula for a scalar function. The fact that you have to integrate $x_1dx_1+x_2dx_2+x_3dx_3$ is a hint that your function is a vector function, so you need to use a slightly different formula. Here is an example of what you need to do.

Answer 2

$x_1=r\cos t$ so $x_1dx_1=r\cos t (-r\sin t )dt $ and so forth. Your integral turns into a real integral of $\int f(t)dt $