Question

How do you find the volume of the region bounded b the line `y = x-2`, the x-axis, `x=2`, and `x=4` is revolved about the line `x = -1`?

Answer 1

`V=(52pi)/3`

Explanation:

`V=pi int_(y_1)^(y_2) (R^2-r^2)dy`

`y=x-2, x=2 => y_1=2-2=0`
`y=x-2, x=4 => y_2=4-2=2`

`x=4 => R=4-(-1)=5`
`y=x-2 => x=y+2 => r=y+2-(-1)=y+3`

`V=pi int_(0)^(2) (5^2-(y+3)^2)dy = pi int_(0)^(2) (25-y^2-6y-9)dy`

`V=pi (16y-3y^2-y^3/3) |_0^2 = pi (32-12-8/3)`

`V=(52pi)/3`