https://math.stackexchange.com/questions/4900918/triple-integration-of-a-volleyball-serve
In[]:=
z==ay^2+by+c/.{{y3,z0},{y0,z2.43},{y-9,z2.02}};cylR=z==ay^2+by+c/.NSolve[%,{}][[1]]//Echo[#,"red cyl."]&;Maximize[%[[2]],y]/.{zz_,{Rule[y,yy_]}}{yyy,zzz};z==ay^2+by+c/.{{y9,z0},%,{y-9,z2.02}};cylG=z==ay^2+by+c/.NSolve[%,{}][[1]]//Echo[#,"green cyl."]&;parR={(y+9)*3/8,y,cylR[[2]]};parG={(y+9)*1/4,y,cylG[[2]]}/.y(3y+9)/2;parsurf=(1-t)parR+tparG;ExpandAll[parsurf]//N//Echo[#,"twisted face (parametric)"]&;netcourt=Graphics3D[{Polygon[{{-4.5,-9,0},{4.5,-9,0},{4.5,9,0},{-4.5,9,0}}],Line[{{-4.5,3,0},{4.5,3,0}}],Line[{{-4.5,-3,0},{4.5,-3,0}}],Line[{{-4.5,0,0},{-4.5,0,2.43}}],Line[{{4.5,0,0},{4.5,0,2.43}}],Polygon[{{-4.5,0,1.43},{4.5,0,1.43},{4.5,0,2.43},{-4.5,0,2.43}}]}];plot1=Show[ContourPlot3D[cylR,{x,-4.5,4.5},{y,-9,6},{z,0,5},RegionFunctionFunction[{x,y},y≥8/3Abs[x]-9],RegionBoundaryStyleNone,ContourStyle{Opacity[0.2],Red},MeshNone,BoundaryStyleNone],ContourPlot3D[cylG,{x,-4.5,4.5},{y,-9,9},{z,0,5},RegionFunctionFunction[{x,y},y≥4Abs[x]-9],RegionBoundaryStyleNone,ContourStyle{Opacity[0.2],Green},MeshNone,BoundaryStyleNone],ParametricPlot3D[parR,{y,-9,3},PlotStyle{Red,Thickness[.0075]}],(*previously:ParametricPlot3D[{(y+9)*1/4,y,paraB[[2]]},{y,-9,9},PlotStyle{Green,Thickness[.0075]}]*)ParametricPlot3D[parG,{y,-9,3},PlotStyle{Green,Thickness[.0075]}],ParametricPlot3D[parsurf,{y,-9,3},{t,0,1},MeshNone],netcourt,PlotRange{{-4.5,4.5},{-9,9},{0,5}},AxesLabel{x,y,z}]
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red cyl.z2.43-0.596111y-0.0712963
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green cyl.z3.81129-0.112222y-0.0345839
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twisted face (parametric){3.375+0.375y,4.5t+y+0.5ty,2.43+0.17597t-0.596111y-0.0391044ty-0.0712963-0.0065174t}
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y
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y
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