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(%);