Syntax:
topfunc and bottomfunc are the two functions (in order)
a, b are the limits of integration (x-values)
the min's and max's are the plot range

topfunc[x_]:=whatever;
bottomfunc[x_]:=whatever;
ShowRevolutionAboutX[topfunc,bottomfunc,{a,b},{{xmin,xmax},{ymin,ymax},{zmin,zmax}}]

Code

ShowRevolutionAboutX[f1_,f2_,{a_,b_},{{xmin_,xmax_},{ymin_,ymax_},{zmin_,zmax_}}]:=Module[{n},
n=36;
Manipulate[
Show[
(*Graphics3D[{
Directive[Lighter[LightGray],Opacity[0.65]],
EdgeForm[None],
Polygon[{{0,0,0},{xmax,0,0},{xmax,0,zmax},{0,0,zmax},{0,0,0}}]}],*)
(*Rotating leader*)
ParametricPlot3D[ 
{x,t f1[x]Cos[ϕ] + (1-t) f2[x]Cos[ϕ],t f1[x]Sin[ϕ] + (1-t) f2[x]Sin[ϕ]},
{t,0,1},{x, a, b},
Mesh->False,
PlotStyle->RGBColor[1,.25,.25]
],
(*Outline of leader*)
Graphics3D[Directive[Thick]],
ParametricPlot3D[ 
{x,f2[x]Cos[ϕ],f2[x]Sin[ϕ]},
{x, a, b},
Mesh->False,
PlotStyle->Green
],
ParametricPlot3D[ 
{x,f1[x]Cos[ϕ],f1[x]Sin[ϕ]},
{x, a, b},
Mesh->False,
PlotStyle->Green
],
Graphics3D[Directive[Thin]],
(*Colored trail drawn behind the leader*)
ParametricPlot3D[ 
{
{x,f1[x]Cos[θ],f1[x]Sin[θ]},
{x,f2[x]Cos[θ],f2[x]Sin[θ]}
},
{x, a, b},
{θ,Pi/2, ϕ},
PlotPoints->20,
PlotStyle->Directive[Lighter[Blue],Opacity[0.65]],
Mesh->False
],
Graphics3D[Line[{{0,0,0},{b,0,0}}]],
Graphics3D[Line[{{0,0,0},{0,b,0}}]],
Graphics3D[Line[{{0,0,0},{0,0,b}}]],
Graphics3D[Text["x",{b+0.05,0,0}]],
Graphics3D[Text["y",{0,0,b+0.05}]],
PlotRange->{{xmin,xmax},{ymin,ymax},{zmin,zmax}},
Boxed->False,
Axes->False
],
{ϕ,Pi/2+0.01,2Pi+Pi/2+0.01, 2π/n}]
];

Examples

f5[x_]:=(E^(-x));
f6[x_]:=0.01;
ShowRevolutionAboutX[f5,f6,{0,1},{{0,1},{-1,1},{-1,1}}]

f1[x_]:=(Sqrt[x]);
f2[x_]:=x;
ShowRevolutionAboutX[f1,f2,{0,Pi},{{0,1},{-1,1},{-1,1}}]

f3[x_]:=Sin[x];
f4[x_]:=0.5 Sin[x];
ShowRevolutionAboutX[f3,f4,{0,Pi},{{0,Pi},{-1,1},{-1,1}}]

f7[x_]:=3+Cos[Pi x/5]
f8[x_]:=0.01;
ShowRevolutionAboutX[f7,f8,{0,20},{{0,20},{-4,4},{-4,4}}]