integration on a parametric curve

Question

Let $\gamma : [0,1] \to \mathbb{S}$ be a parametric curve, which $\mathbb{S}$ is the unit circle in the plane. I want to integrate the function $f= f(r , \theta ) $ on the $\Gamma = \gamma ([0,1])$, i.e. I want to calculate $\int _{\Gamma} f$.

What is the correct formulation of the problem? (We know that $\gamma (0)$ and $\gamma (1) $ lies on the boundary of $\mathbb S$).

Answer 1

If $r\in(0,\infty)$ and $\theta\in(-\pi,\pi)$ it should be

$$\int_{\Gamma}f=\int_0^1f(|\gamma(t)|,\arg(\gamma (t)))|\gamma'(t)|dt$$