On 8/1/23, Undescribed Horrific Abuse, One Victim & Survivor of Many <[email protected]> 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
