If a curve has two different parametrizations, say $\gamma$ and $\beta$, then $\int_{\gamma}f(z) dz=\int_{\beta}f(z)dz?$

Question

If a curve has two different parametrizations, say $\gamma$ and $\beta$, then

$$\int_{\gamma}f(z) dz=\int_{\beta}f(z)dz?$$

I think the proposition is true. My idea is showing that the hypotesis implies that $\gamma$ is a reparametrization of $\beta$ and I already know that, in this case, we have the equality.

Please, tell me if my analysis is true. Or could I find A counterexample?

Here, $\int_{\gamma}f(z) dz=\int_{a}^{b} f(\gamma(t))\gamma'(t) dt$ and $a\le t\le a$.

Answer 1

We know that $\int_{\gamma}f(z)dz=-\int_{\gamma^-}f(z)dz$ where $\gamma^-(t):=\gamma(a+b-t)$ and $\gamma^-$ and $\gamma$ parameterize the same curve, so this is not generally true.