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

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

wedge_volume = integral(triangle_area, x=-r_sphere..+r_sphere)
triangle_area = 1/2 r_slice^2 dtheta
r_slice = sqrt(r_sphere^2 - x^2)
wedge_volume = I{-r_sphere..+ triangle_area dx
wedge_volume = I{-r_sphere..+ 1/2 r_slice^2 dtheta dx
wedge_volume = I{-r_sphere..+ 1/2 sqrt(r_sphere^2 - x^2)^2 dtheta dx
wedge_volume = I{-r_sphere..+ 1/2 (r_sphere^2 - x^2) dtheta dx

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

next step: integrate (r_sphere^2 - x^2) dx