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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.
Solution
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.
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