Question

How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations `y=e^-x`, x= -1, x = 2 and the x-axis are rotated about the x-axis?

Answer 1

`VOR = pi/2 (e^2-e^(-4))`
`" " ~~11.5779320...`

Explanation:

I recommend that you always draw a sketch to clarify what needs calculating.

graph{e^(-x) [-2, 3, -5, 5]}

The Volume of Revolution about `Ox` is given by:

`VOR = int_alpha^beta \ piy^2 \ dx`

So in this case:

`VOR = int_-1^2 \ pi \ (e^-x)^2 \ dx`
`" " = pi \ int_-1^2 \ e^(-2x) \ dx`
`" " = pi [ -1/(2) e^(-2x) ]_-1^2`
`" " = -pi/2 {(e^(-4))-(e^2)}`
`" " = pi/2 (e^2-e^(-4))`
`" " ~~11.5779320...`