a wedge made of a stack of triangles, each one with area of 1/2 r^2 dtheta where r is the radius of a sphere slice uhh so that's really sqrt(r^2 - x) where x runs from -r to +r
two different r's here let's subscript them a wedge made of a stack of triangles each one with area of 1/2 r_slice^2 dtheta now, r_slice^2 + height^2 = r_sphere^2
cognitive maintenance habit looks like the problem happened after this point
so if we run x=height from -r to r r_slice = sqrt(r_sphere^2 - x^2)
then maybe plug in to other expression [somewhere near right here we failed to include important information; forgot something important] [the x=height line should have retained information regarding the goal of integration, for the dvolume line below. it did not. i collect informaiton on the problem but it is hard to heal.]
dvolume = 1/2 sqrt(r_sphere^2 - x^2)^2 dtheta
and before this point
important to drill down and find the choice that needs [more neuron/decisions?] {we think a little maybe in terms of "hooks" and "triggers" --
dvolume = 1/2(r_sphere^2 - x^2) dtheta
maybe unsure
holy frack i just wrote dvolume when i still had only plugged in for triangle area then seemingly misinterpreted my writing as trying to integrate something when i was just preparing to still and possibly worked super hard to inform my mind to improve around internal events that may not have happened because things i was looking at, and had just done, were hidden to me