When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical cap by the following integral:
$$\int\limits_{y_1}^{y_2} {\int\limits_{-\sqrt {1 - {y^2}-{z_1^2}} }^{\sqrt {1 - {y^2}-{z_1^2}} }{\sqrt {1 - {x^2}-{y^2}} } -z_1\,dx\,dy}$$
This is something Wolfram alpha can calculate. Now I want to take this one step further and find the volumes when the spherical cap is cut by a plane that is not perpendicular on $z=z_1$, but makes an angle between $0$ and $90°$.
The question boils down to calculating the volume of the orange region.
I can calculate the total volume left of the vertical plane with the integral above. I only have to substract the orange region.
Suppose the angle between the left face (with the yellow normal vector on it) of this region makes an angle $\alpha$ with plane $y=y_1$. It looks like a spherical wedge however this one doesn't start in the origin of the sphere. I can't figure out how to set up the integral to calculate this orange volume. Does anyone have an idea?