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Question
Find the volume of the solid generated by the complete revolution of the ellipse `"x"^2/36 + "y"^2/25 = 1` about Y-axis.
Solution
The equation of the ellipse is
`"x"^2/36 + "y"^2/25 = 1`
i.e. `"x"^2/36 = 1 - "y"^2/25 `
`therefore "x"^2 = 36/25 (25 - "y"^2)`
Let V be the required volume of the. solid obtained by revolving the ellipse about Y-axis.
`therefore "V" = pi int_-5^5 "x"^2 "dy"`
`= pi int_-5^5 36/25 (25 - "y"^2) "dy"`
`= 36/25 xx pi xx 2 int _0^5 (25 - "y"^2) "dy"`
....`[because int_(-"a")^"a" "f(x)" "dx" = 2 int_0^"a" "f(x)" "dx"]`
`= (72 pi)/25 [25"y" - "y"^3/3]_0^5`
`= (72 pi)/25 [25(5) - 5^3/3 - 0]`
`= (72 pi)/25 [125 - 125/3] = (72 pi)/25 [250/3]`
= 240 π cubic units.
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