Shells versus Washers and Discs:
Section 5.3
Andy Long

#8, p. 367

f1[x_]:=Sqrt[x]
f2[x_]:=x^2
Plot[{f1[x],f2[x]},{x,0,1}]
Plot[{f1[Abs[x]],f2[x]},{x,-1,1}]
(* 
RegionPlot3D[-1 ≤ x≤1 && y ≤ Sqrt[Abs[x]] && y≥x^2,{x,-1,1},{y,-1,1},{z,0,1}] 
*)

Cylindrical Shells:

Integrate[2 Pi x (f1[x]-f2[x]),{x,0,1}]

Washers:

g1[y_]:=y^2
g2[y_]:=Sqrt[y]
Integrate[Pi (g2[y]^2-g1[y]^2),{y,0,1}]