<<< 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") ] ] ] ] >>> tex S = \{ 0 \leq \phi \leq 2\pi,\ 0 \leq \theta \leq \pi,\ 0 \leq \rho \leq R\}\begin{aligned} \mathrm{Volume} & = \iiint\limits_{S}\!\rho^{2}\sin\theta\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi \\ & = \int_{0}^{2\pi}\!\mathrm{d}\phi\,\int_{0}^{\pi}\!\sin\theta\,\mathrm{d}\theta\,\int_{0}^{R}\!\rho^{2}\mathrm{d}\rho \\ & = \phi\Big|_{0}^{2\pi}\ \left( -\cos\theta \right)\Big|_{0}^{\pi}\ \tfrac{1}{3}\rho^{3}\Big|_{0}^{R} \\ & = 2\pi \times 2 \times \tfrac{1}{3}R^{3} \\ & = \tfrac{4}{3}\pi R^{3} \end{aligned}