Question

Y=x, y=0, y=4, x=6 : find the volume of the solid generated revolving around line x=6

Answer 1

Using shells, the volume is


`\displaystyle2\pi\int_0^6(6-x)x\,\mathrm dx`

where
`6-x` is the distance from any point in the region along the x-axis to the axis of revolution
`x=6`, which makes up the radius of each shell; and
`x` is the height of each shell, which is determined by the line
`y=x`.

So the volume is


`\displaystyle2\pi\int_0^6(6x-x^2)\,\mathrm dx=2\pi\left(3x^2-\frac{x^3}3\right)\bigg|_(x=0)^(x=6)=72\pi`