For
NIntegrate[Sqrt[1 + x^2] + y^2 - z^4, {x, y, z} \[Element] Ball[]]
a change of variables is made that remaps it to an integral over the unit cube, which is integrated with the "MultiDimensionalRule":
NIntegrate[
8 Sqrt[1 - x^2] Sqrt[1 - x^2 - (1 - x^2) y^2] (Sqrt[1 + x^2] + (1 - x^2) y^2 - (1 - x^2 - (1 - x^2) y^2)^2 z^4),
{x, 0, 1}, {y, 0, 1}, {z, 0, 1}]
For a discretized mesh region, NIntegrate calls an FEM integrator that integrates over each mesh element using integration rules for each element type.
