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

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