[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) ]

this looks reasonable r_sphere^2 >= x^2

>
> indefinite integral of r_sphere^2 dx = r_sphere^2 * x
> indefinite integral of x^2 dx = 2x^3

noting that both of these have x raised to an odd power (a little hard
to notice given the location of the ^2's). because x is raised to an
odd power in both, they will both have the negative and positive parts
of the integral in the same relation, and their inequality relation
should be preserved. [at first i thought one had an even power, and
then imagining plugging in made it not preserved]

> evaulate fo rdomain
> r_sphere^2 * r_sphere - r_sphere^2 * -r_sphere = 2rsphere^3
this above line looks reasonably likely to be right
> 2(r_sphere)^3 - 2(-r_sphere)^3 = 4 r_sphere^3
here the errormistake is findable. the above two lines reveal it. i
haven't found it yet
>
> 2rsphere^3 - 4 r_sphere^3 = -2r_sphere^3 [mistake indicated --
>