[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
>
> here are parts: r^2 > x^2
> we expect when integrating dx -r..+r for the relation to hold.
> what changed?
> integral r^2 dx = 2r^2 x
> ok there's one mistake i had dropped that 2
> integral x^2 dx = 3x^2
here it is, this is 3x^3 not 3x^2
> let's subtract before plugging
> 2r^2 x - 3x^2 | x=-r..+r
2r^2 x - 3x^3 | x=-r..+r
> ohh then i can handle positive and negative separately
> for +r: 2(r^2)r - 3(r^2) = 2r^3 -  -- mistake indicated. [parts should
> match in exponent

+r: 2(r^2)r - 3(r^3) = 2r^3 - 3r^3 negative.
is there a reason for this to be negative? it represents a partial area, right?