Question

What is the Volume of Revolution if the area bounded by the curve `y=x^2-4x` and the `x`-axis is is rotated about the `x`-axis?

Answer 1

`(512 pi) /15 \ unit^3`

Explanation:

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

graph{x^2-4x [-10, 10, -5, 5]}

The curve intersects the `x`-axis, when:

`x^2-4x = 0 => x(x-4)=0 => x=0,4`

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

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

So in this case:

`VOR = int_0^4 \ pi \ (x^2-4x)^2 \ dx`
`" " = pi \ int_0^4 \ (x^4-8x^3+16x^2) \ dx`
`" " = pi [ x^5/5-2x^4+16x^3/3]_0^4`
`" " = pi { (4^5/5-2*4^4+16*4^3/3)-(0) }`
`" " = pi (1024/5-512+1024/3)-(0) }`
`" " = (512 pi) /15`