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 [i wrote the sqrt part this. this was difficult. then i added dvolume to try to figure out what i was doing. taking notes seems very weird, but it did help me find this similarity here. the behavior of relaxing and focusing on a small part of a puzzle when experiencing stress, does not always retain the outer context for doing the small part ... [that's a skill i could build
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
oh … i had actually narrowed in on the event of writing dvolume wrongly, which was indeed the mistake … !