Hi all,

I'm trying to minimize the following integral : $ \int_{0}^{\pi/2} \frac{\int_{0}^{x} \sqrt{r(\theta)^2 + r'(\theta)^2}d\theta}{\sin(x)} dx $ with boundary values r(0)=1 and r(pi/2)=0. As you may have guessed, the numerator in the integrand represents arc length in polar coordinates of the curve $(\theta, r(\theta) )$. I have absolutely no idea where to start: I have tried looking into optimization and variational calculus books but wasn't lucky. Are there numerical methods which I could try? Of course, an extra requirement is that $ \underset{x\to 0^{+}}{\lim} \frac{\int_{0}^{x} \sqrt{r(\theta)^2 + r'(\theta)^2}d\theta}{\sin(x)}$ exists and is bounded.