Question

Find the volume of the solid obtained by revolving about the X-axis, the region bounded by the curve `"x"^2/4 - "y"^2/9 = 1` and the lines x = 2 , x = 4.

Answer 1

Given equation of the curve is 

`"x"^2/4 - "y"^2/9 = 1` , which is a hyperbola

`therefore "y"^2/9 = "x"^2/4 - 1` 

`therefore "y"^2 = 9/4 ("x"^2 - 4)`

`therefore  "Volume" = pi int_2^4 "y"^2  "dx"`

`= (9pi)/4 int _2^4 ("x"^2 - 4) "dx"`

`therefore "V" = (9pi)/4 ["x"^3/3 - 4"x"]_2^4`

`= (9pi)/4 [(64/3 - 16) - (8/3 - 8)]`

= 24 π cubic units.