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.