Cylinder volumes

In any d-dimensional space, the volume of a cylinder of length x and radius r whose direction is defined by a unit vector v turns out to be given by

s[d-1] r^(d-1) x (1 - (d-1)/(d+1)(RicciScalar-RicciTensor . v .v) r^{2} + …)

Note that what determines the volume of the cylinder is curvature orthogonal to its direction—and this is what leads to the combination of Ricci scalar and tensor that appears.