Consider the function $r=f(\theta)$ in polar coordinates. The length of an arc of a circle is just $$S=\theta r$$ Where $r$ is the radius of the circle and $\theta$ is the angle that represents this arc. But since $r=f(\theta)$ and $\theta$ Should approach zero so that we can get the exact value of the arc, So $$\,dS=f(\theta)\,d\theta$$ Integrating from $\theta_1$ to $\theta_2$, we get : $$S=\int_{\theta_1}^{\theta_2} f(\theta)\,d\theta $$ But the actual formula for the length of a curve in polar coordinates is $$\int_{\theta_1}^{\theta_2} \sqrt{f^2(\theta)+f’(\theta)^2}\,d\theta.$$
I know that my approach isn’t rigorous enough, but it’s is still reasonable, so why it is different from the actual formula?
Your equation for $dS$ is incorrect. In polar coordinates the differential length would be $f(\theta) d\theta$ only if $r$ is a constant. If you draw a little diagram you'll see that, in fact, the differential element of length is given by
$$dS=\sqrt{f^2+ \bigg(\frac{df}{d\theta}\bigg)^2}\ d\theta$$