[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
evaulate fo rdomain r_sphere^2 * r_sphere - r_sphere^2 * -r_sphere = 2rsphere^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] 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 --