Pre-Requisite for My Problem
There's this cardioid curve and I need to find it's perimeter. The equation given by my teacher is
$$R=a(1+\cos\theta)$$
Here,
$R$ : the distance to and part of the curve from the origin.
$\theta : $ the angle made by $R$ w. r. t. positive x-axis.
He said when we are given an equation like $r=f(\theta)$, then in order to obtain the perimeter of the desired curve we need to apply this formula.
$$S=\int_{\alpha}^{\beta}\sqrt{\bigg(r^2+\big(\dfrac{dr}{d\theta}\big)^2\bigg)}\space d\theta$$
where
$S :$ perimeter of the curve
$\alpha:$ initial angle $R$ makes with positive x-axis
$\beta:$ final angle $R$ makes with positive x-axis
Now, when I apply all these to find the perimeter of the cardioid where $\theta$ ranges from $0$ to $2\pi$.
My Problem
Now, I am really banging my head for the integration part. It turns out that when you integrate from $0$ to $2\pi$, the result is $ZERO!$. But if we apply the Property of Definite Integral and multiply $2$ with the whole integration and then integrate the function from $0$ to $\pi$, then we get the result as $8a$.
Can anyone please point out the mistake in my approach?