<<< native [ EIdentifier "S" , ESymbol Rel "=" , ESymbol Open "{" , ENumber "0" , ESymbol Rel "\8804" , EIdentifier "\981" , ESymbol Rel "\8804" , ENumber "2" , EIdentifier "\960" , ESymbol Pun "," , ESpace (2 % 9) , ENumber "0" , ESymbol Rel "\8804" , EIdentifier "\952" , ESymbol Rel "\8804" , EIdentifier "\960" , ESymbol Pun "," , ESpace (2 % 9) , ENumber "0" , ESymbol Rel "\8804" , EIdentifier "\961" , ESymbol Rel "\8804" , EIdentifier "R" , ESymbol Close "}" , EArray [ AlignRight , AlignLeft ] [ [ [ EStyled TextNormal [ EIdentifier "V" , EIdentifier "o" , EIdentifier "l" , EIdentifier "u" , EIdentifier "m" , EIdentifier "e" ] ] , [ ESymbol Rel "=" , EUnder False (ESymbol Op "\8749") (EIdentifier "S") , ESpace ((-1) % 6) , ESuper (EIdentifier "\961") (ENumber "2") , EMathOperator "sin" , EIdentifier "\952" , ESpace (1 % 6) , EStyled TextNormal [ EIdentifier "d" ] , EIdentifier "\961" , ESpace (1 % 6) , EStyled TextNormal [ EIdentifier "d" ] , EIdentifier "\952" , ESpace (1 % 6) , EStyled TextNormal [ EIdentifier "d" ] , EIdentifier "\981" ] ] , [ [] , [ ESymbol Rel "=" , ESubsup (ESymbol Op "\8747") (ENumber "0") (EGrouped [ ENumber "2" , EIdentifier "\960" ]) , ESpace ((-1) % 6) , EStyled TextNormal [ EIdentifier "d" ] , EIdentifier "\981" , ESpace (1 % 6) , ESubsup (ESymbol Op "\8747") (ENumber "0") (EIdentifier "\960") , ESpace ((-1) % 6) , EMathOperator "sin" , EIdentifier "\952" , ESpace (1 % 6) , EStyled TextNormal [ EIdentifier "d" ] , EIdentifier "\952" , ESpace (1 % 6) , ESubsup (ESymbol Op "\8747") (ENumber "0") (EIdentifier "R") , ESpace ((-1) % 6) , ESuper (EIdentifier "\961") (ENumber "2") , EStyled TextNormal [ EIdentifier "d" ] , EIdentifier "\961" ] ] , [ [] , [ ESymbol Rel "=" , EIdentifier "\981" , ESubsup (EScaled (9 % 5) (ESymbol Open "|")) (ENumber "0") (EGrouped [ ENumber "2" , EIdentifier "\960" ]) , ESpace (2 % 9) , EDelimited "(" ")" [ Right (ESymbol Op "\8722") , Right (EMathOperator "cos") , Right (EIdentifier "\952") ] , ESubsup (EScaled (9 % 5) (ESymbol Open "|")) (ENumber "0") (EIdentifier "\960") , ESpace (2 % 9) , EFraction InlineFrac (ENumber "1") (ENumber "3") , ESuper (EIdentifier "\961") (ENumber "3") , ESubsup (EScaled (9 % 5) (ESymbol Open "|")) (ENumber "0") (EIdentifier "R") ] ] , [ [] , [ ESymbol Rel "=" , ENumber "2" , EIdentifier "\960" , ESymbol Bin "\215" , ENumber "2" , ESymbol Bin "\215" , EFraction InlineFrac (ENumber "1") (ENumber "3") , ESuper (EIdentifier "R") (ENumber "3") ] ] , [ [] , [ ESymbol Rel "=" , EFraction InlineFrac (ENumber "4") (ENumber "3") , EIdentifier "\960" , ESuper (EIdentifier "R") (ENumber "3") ] ] ] ] >>> 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