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.

Answer 1

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.