For point P(x,y) the radius of the cylindrical shell is 3-x and the thickness of the cylindrical wall is dx. The height of the cylinder is y=9x-x2. The volume of the shell is 2π(3-x)(9x-x2)dx, because the volume of a cylindrical shell is 2πrht where r=radius=3-x, h=height=y and t=thickness=dx.
The volume of the solid is the sum of all the shells, which gives us the integral: 2π∫x(3-x)(9-x)dx.
2π∫(27x-12x2+x3)dx for 0≤x≤3=2π[27x2/2-4x3+x4/4]03.
Inserting the limits we get: 135π/2=212.058 cubic units approx.