Shells

Find the volume of the solid obtained by rotating about the y-axis the region bounded by y=x(x-2)^2 and y=2.

The region to be rotated is

> restart;

> with (plots):

> c1:=y=x*(x-2)^2;

> c2:=y=2;

> with(plots):

> implicitplot({c1,c2},x=0..3,y=-3..3);

From the graph, we see a point of intersection near x=2.8

> subs(c2,c1);

> UB:=fsolve(%,x);

The volume is given by

> Int(2*Pi*x*(2-x*(x-2)^2),x=0..UB);

> value(%);