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

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

On 8/1/23, Undescribed Horrific Abuse, One Victim & Survivor of Many
<gmkarl at gmail.com> wrote:
>> [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
let's subtract before plugging
2r^2 x - 3x^2 | 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