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由对称性， 得
V = 2∫<0, 2π>dt∫<0, Rcost> √(R^2-r^2)rdr
= -∫<0, 2π>dt∫<0, Rcost> √(R^2-r^2)d(R^2-r^2)
= -∫<0, 2π>dt[(2/3)(R^2-r^2)^(3/2)]<0, Rcost>
= (2R^3/3)∫<0, 2π>[1-(sint)^3]dt
= (2R^3/3){2π+∫<0, 2π>[1-(cost)^2]dcost}
= (2R^3/3){2π+[cost-(1/3)(cost)^3]<0, 2π>}
= (2R^3/3){2π+[cost-(1/3)(cost)^3]<0, 2π>} = (4/3)πR^3
V = 2∫<0, 2π>dt∫<0, Rcost> √(R^2-r^2)rdr
= -∫<0, 2π>dt∫<0, Rcost> √(R^2-r^2)d(R^2-r^2)
= -∫<0, 2π>dt[(2/3)(R^2-r^2)^(3/2)]<0, Rcost>
= (2R^3/3)∫<0, 2π>[1-(sint)^3]dt
= (2R^3/3){2π+∫<0, 2π>[1-(cost)^2]dcost}
= (2R^3/3){2π+[cost-(1/3)(cost)^3]<0, 2π>}
= (2R^3/3){2π+[cost-(1/3)(cost)^3]<0, 2π>} = (4/3)πR^3
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