How Do You Calculate the Intersection of Two Spheres in 3D Integrals?

[kassem84]

Hello,
I am calculating some integrals in 3 dimensions. However, the difficulties of such integrals lie in the determination of the boundaries of the variables integrated over.

\int_{C} d^{3}\vec{t} e^{-\vec{s}.\vec{t}}
For example, if we consider (C) as the region of the intersection of 2 spheres:
C=|\vec{s}-\vec{t}|<1 and |\vec{s}+\vec{t}|<1
I want to solve these set of inequalities for fixed \vec{s}, using spherical coordinates.
i.e. determine the interval over |\vec{t}|, \phi and \vartheta=angle(\vec{s},\vec{t})

Does anyone have a strategy to deal with such inequalities?

Thanks in advance.^{}

 

[Simon Bridge]

Use cylindrical-polar coordinates with the z axis perpendicular to the plane of intersection.
You can treat it as a volume of rotation.