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