[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
good, 1 in denominator
indefinite integral of x^2 dx = 2x^3 wrong, used derivative rule instead of integration rule. I(x^2,dx) = x^3/3 = (1/3) x^3 . can check rule by deriving area of triangle. it really should be (1/3) x^3, see derivation of triangle area. it is more right than 2x^3 for sure.
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 using (1/3)x^3, (1/3)r_sphere^3 - (1/3)(-r_sphere)^3 = 2/3 r_sphere^3 :D 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 r^2 dx => r^2 x . the rule and denominator comes from the variable of integration, not the coefficient. the rule is to divide, not multiply as in differentiation. r^2 x is correct. ok there's one mistake i had dropped that 2 this was not the mistake, which was a further mistake [thinking some on dissociative delusion, seems there was one here integral x^2 dx = 3x^2 here it is, this is 3x^3 not 3x^2 this is (1/3)x^3 as described above :) see derivation of triangle area to support rule let's subtract before plugging 2r^2 x - 3x^2 | x=-r..+r 2r^2 x - 3x^3 | x=-r..+r r^2 x - (1/3)x^3 | x=-r..+r (note this is already done once above) 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? +r: r^2 r - (1/3)r^3 = r^3 - (1/3) r^3 = (2/3) r^3 -r: r^2 (-r) - (1/3)(-r)^3 = -r^3 + (1/3)r^3 = -(2/3) r^3 difference = 4/3 r^3 then finish other avenue where the parts were subtracted separate and then summed