fractionalIntegral[alpha_,x_]:=1/Gamma[alpha]Integrate[(x-t)^(alpha-1)t^2,{t,0,x}]

fractionalFirstDeriv[v_,x_]:=Derivative[0,1][fractionalIntegral][1-v,x]
fractionalSecondDeriv[v_,x_]:=Derivative[0,2][fractionalIntegral][2-v,x]

fractionalFirstDeriv[v,s]
fractionalSecondDeriv[v,s]

(* Note these 2 expressions are the same after using properties of the Gamma function to simplify. No need for a piecewise function to have a nice form for derivatives between 0 and 2. *)

tubeThickness:=0.05
tubeStretchFactor:=10

goodForm[v_,x_]:=2x^(2-v)(3-v)/Gamma[4-v]
goodPlot:=Plot3D[goodForm[2-v,t],{v,0,2},{t,1,7},PlotTheme->"FilledSurface",PlotRange->{0,50},MaxRecursion->6]

constantPlot:=Plot3D[2+tubeStretchFactor*Sqrt[(tubeThickness/2)^2-x^2],{x,0,tubeThickness/2},{y,1,7},PlotTheme->"FilledSurface"]
linearPlot:=Plot3D[2*y+tubeStretchFactor*Sqrt[(tubeThickness/2)^2-(x-1)^2],{x,1-tubeThickness/2,1+tubeThickness/2},{y,1,7},PlotTheme->"FilledSurface"]
quadraticPlot:=Plot3D[y^2+tubeStretchFactor*Sqrt[(tubeThickness/2)^2-(x-2)^2],{x,2-tubeThickness/2,2},{y,1,7},PlotTheme->"FilledSurface"]

output=Show[goodPlot,constantPlot,linearPlot,quadraticPlot,PlotRange->{0,50}]
Export[NotebookDirectory[]<>"Fractional Quadratic.stl",output]