sphere_volume.test

<<< native
[ EIdentifier "S"
, ESymbol Rel "="
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, ESymbol Rel "\8804"
, EIdentifier "\981"
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>>> typst
S = { 0 lt.eq phi.alt lt.eq 2 pi\,med 0 lt.eq theta lt.eq pi\,med 0 lt.eq rho lt.eq R } upright(V o l u m e) & = integral.triple_S #h(-1em) rho^2 sin theta thin upright(d) rho thin upright(d) theta thin upright(d) phi.alt\
 & = integral_0^(2 pi) #h(-1em) upright(d) phi.alt thin integral_0^pi #h(-1em) sin theta thin upright(d) theta thin integral_0^R #h(-1em) rho^2 upright(d) rho\
 & = phi.alt #scale(x: 180%, y: 180%)[\|]_0^(2 pi) med (- cos theta) #scale(x: 180%, y: 180%)[\|]_0^pi med 1 / 3 rho^3 #scale(x: 180%, y: 180%)[\|]_0^R\
 & = 2 pi times 2 times 1 / 3 R^3\
 & = 4 / 3 pi R^3