sphere_volume.test

<<< 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")
        ]
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    , [ []
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        , ENumber "2"
        , EIdentifier "\960"
        , ESymbol Bin "\215"
        , ENumber "2"
        , ESymbol Bin "\215"
        , EFraction InlineFrac (ENumber "1") (ENumber "3")
        , ESuper (EIdentifier "R") (ENumber "3")
        ]
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    , [ []
      , [ 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 ( \[u2212] 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} }
}