Based (copying) on andre'sandre's answer here is a modification.
replaced http://mathematica.stackexchange.com/ with https://mathematica.stackexchange.com/
face = Import["http://cliparts.co/cliparts/6Ty/ogn/6TyognE8c.png"] 
cow = Graphics[{Disk[10 {RandomReal[], RandomReal[]}, RandomReal[]] & /@ Range[20], Inset[face]}, AspectRatio -> 1,ImageSize -> 500]; ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], Cos[v]}, {u, 0, 2 Pi}, {v, 0, Pi}, Mesh -> None, PlotPoints -> 100, TextureCoordinateFunction -> ({#4, 1 - #5} &), Boxed -> False, PlotStyle -> Texture[Show[cow, ImageSize -> 1000]], Lighting -> "Neutral", Axes -> False, RotationAction -> "Clip"] 
face = Import["http://cliparts.co/cliparts/6Ty/ogn/6TyognE8c.png"]
cow = Graphics[{Disk[10 {RandomReal[], RandomReal[]}, RandomReal[]] & /@ Range[20], Inset[face]}, AspectRatio -> 1,ImageSize -> 500]; ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], Cos[v]}, {u, 0, 2 Pi}, {v, 0, Pi}, Mesh -> None, PlotPoints -> 100, TextureCoordinateFunction -> ({#4, 1 - #5} &), Boxed -> False, PlotStyle -> Texture[Show[cow, ImageSize -> 1000]], Lighting -> "Neutral", Axes -> False, RotationAction -> "Clip"]
face = Import["http://cliparts.co/cliparts/6Ty/ogn/6TyognE8c.png"] cow = Graphics[{Disk[10 {RandomReal[], RandomReal[]}, RandomReal[]] & /@ Range[20], Inset[face]}, AspectRatio -> 1,ImageSize -> 500]; ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], Cos[v]}, {u, 0, 2 Pi}, {v, 0, Pi}, Mesh -> None, PlotPoints -> 100, TextureCoordinateFunction -> ({#4, 1 - #5} &), Boxed -> False, PlotStyle -> Texture[Show[cow, ImageSize -> 1000]], Lighting -> "Neutral", Axes -> False, RotationAction -> "Clip"]
Table[Show[cowTable[vcow = NIntegrate[1, {x, y, z} ∈ MeshRegion[(# ((Norm[#]/Rcow)^-coeff)) & /@ cow[[1, 2, 1]], cow[[1, 2, 2]]]]; Show[cow /. GraphicsComplex[array1_, rest___] :> GraphicsComplex[(# ((Norm[#]/Rcow)^-coeff)) & /@ array1, rest], Axes -> True, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}} 0.6, Boxed -> True, PlotLabel -> Framed[coeff]StringForm["(``), V=``", coeff, vcow], ImageSize -> 200], {coeff, 0, 1, 0.25}] 
NIntegrate[1,{x,y,z}∈MeshRegion[(#((Norm[#]/Rcow)^-1))&/@cow[[1,2,1]], cow[[1,2,2]]]] 3.88847
Although the final radius is same as Rcow, You do not get Vcow for the volume keeps increasing because, on this sphere, several layers are overlapping on each other (reminds me the length of British coastline) which causes overcounting during the numerical integration.
Table[Show[cow /. GraphicsComplex[array1_, rest___] :> GraphicsComplex[(# ((Norm[#]/Rcow)^-coeff)) & /@ array1, rest], Axes -> True, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}} 0.6, Boxed -> True, PlotLabel -> Framed[coeff], ImageSize -> 200], {coeff, 0, 1, 0.25}]
NIntegrate[1,{x,y,z}∈MeshRegion[(#((Norm[#]/Rcow)^-1))&/@cow[[1,2,1]], cow[[1,2,2]]]] 3.88847
Although the final radius is same as Rcow, You do not get Vcow for the volume because, on this sphere, several layers are overlapping on each other (reminds me the length of British coastline).
Table[vcow = NIntegrate[1, {x, y, z} ∈ MeshRegion[(# ((Norm[#]/Rcow)^-coeff)) & /@ cow[[1, 2, 1]], cow[[1, 2, 2]]]]; Show[cow /. GraphicsComplex[array1_, rest___] :> GraphicsComplex[(# ((Norm[#]/Rcow)^-coeff)) & /@ array1, rest], Axes -> True, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}} 0.6, Boxed -> True, PlotLabel -> StringForm["(``), V=``", coeff, vcow], ImageSize -> 200], {coeff, 0, 1, 0.25}]
Although the final radius is same as Rcow, the volume keeps increasing because, on this sphere, several layers are overlapping on each other (reminds me the length of British coastline) which causes overcounting during the numerical integration.