<<< 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") ] ] ] ] >>> eqn S = \[lC] 0 <= phi <= 2 pi , ~ 0 <= theta <= pi , ~ 0 <= rho <= R \[rC] matrix{ rcol{ {roman {V o l u m e}} above {""} above {""} above {""} above {""} } lcol{ {= \[u222D] from S back 16 rho sup 2 sin theta ^ roman {d} rho ^ roman {d} theta ^ roman {d} phi} above {= int sub 0 sup {2 pi} back 16 roman {d} phi ^ int sub 0 sup pi back 16 sin theta ^ roman {d} theta ^ int sub 0 sup R back 16 rho sup 2 roman {d} rho} above {= phi | sub 0 sup {2 pi} ~ left ( - cos theta right ) | sub 0 sup pi ~ 1 over 3 rho sup 3 | sub 0 sup R} above {= 2 pi times 2 times 1 over 3 R sup 3} above {= 4 over 3 pi R sup 3} } }