All Questions

Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...

Is it (for some reason) true that $\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt=\frac{\pi}{2}$? Numerical evidence (from Mathematica): when $c=1/5$, the integral is $\...