[ot][ot][ot][personal] whirligig small thoughts

[Undescribed Horrific Abuse, One Victim & Survivor of Many]

[if output is wrong, derivation of wedge integration expression is
pending check from start]

wedge_volume = dtheta/2 I{-r_sphere..+ (r_sphere^2 - x^2) dx

integration domain: x=(-r_sphere, +r_sphere)
wedge_volume = dtheta/2 [ integrate(r_sphere^2) - integrate(x^2) ]

indefinite integral of r_sphere^2 dx = r_sphere^2 * x
indefinite integral of x^2 dx = 2x^3
evaulate fo rdomain
r_sphere^2 * r_sphere - r_sphere^2 * -r_sphere = 2rsphere^3
2(r_sphere)^3 - 2(-r_sphere)^3 = 4 r_sphere^3

2rsphere^3 - 4 r_sphere^3 = -2r_sphere^3 [mistake indicated --