What is the volume of this region:
$ C(x,\eta(x),\theta,r) = \{{x+\lambda y\in\mathbb{R}^d: y\in\mathbb{R}^d, ||y||_2=1, y^T\eta(x)\geq \cos\theta, \lambda\in[0,r]\}} $
where $\theta\in(0,\pi/2)$, $r>0$ and $\eta(x)$ is a unit vector. This is a rather abstract way of writing down a cone with a cap (but also in higher dimensions).
For instance, for $d=1$ (then $\theta=0$) the region is a line of length $r$.
For $d=2$ the region is the segment of a circle with area $\frac{\theta}{\pi}\pi r^2 = \theta r^2$.
I'm after a general formula for $d\geq 2$. A first guess would be
$ Vol(C(x,\eta(x),\theta,r)) = \left(\frac{\theta}{\pi}\right)^{d-1}\sigma_d r^d $
where $\sigma_d$ is the volume of the $d$-dimensional unit ball. Can anyone show this?