x=3 and y=9x-x4/12 meet when y=9×3-81/12=27-27/4=81/4.
The distance between any point P(x,y) on the curve and the line forms the radius of a circle and a disc is created by the rotating point. The area of the disc is π(3-x)2. If the disc has a thickness dy, its volume is π(3-x)2dy so the volume of the solid created by many stacked discs=0∫81/4π(3-x)2dy. Since y=9x-x4/12, dy can be replaced by (9-x3/3)dx, and the limits changed to 0 (low) and 3 (high) for x.
We can now evaluate the integral:
π0∫3(3-x)2(9-x3/3)dx=
π0∫3(9-6x+x2)(9-x3/3)dx=
π0∫3(81-54x+9x2-3x3+2x4-x5/3)dx=
π[81x-27x2+3x3-3x4/4+2x5/5-x6/18]03=
π(243-243+81-243/4+486/5-81/2)=1539π/20=76.95π=241.75 cubic units approx.