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get the mathematical expression for the volume formula. known coordinate system x, y, z receive .stl model enter image description here enter image description here

rb1 = 28.5;
r00 = 32;
PhiEnd = 0.318;
PsiEnd = 0.6144;
(***********************************)
x = rb1*Sin[Psi] - (rb1*Psi - r00*(1 - Cos[Phi]))*Cos[Psi]; 
y = rb1*Cos[Psi] + (rb1*Psi - r00*(1 - Cos[Phi]))*Sin[Psi];
z = r00*Sin[Phi];
(***********************************)
func1 = {x, y, z};
func = {{x, y, z}, {Phi, 0, PhiEnd}, {Psi, 0, 
PsiEnd}}; 
(*a=ParametricPlot3D[{x, y,z}, {Phi, 0, PhiEnd}, {Psi, \
0,PsiEnd},MaxRecursion\[Rule]4,PlotStyle\[Rule]{Orange,Specularity[\
White,10]},Axes\[Rule]None,Mesh\[Rule]None]*)
a = ParametricPlot3D @@ func ;(*Parametric function definition*)
(***********************************)
Off[NIntegrate::slwcon]; (*Disable warning when \
integration*)
(***********************************)
ar = Area @@ 
  func ;(*Calculation of surface area embedded numerical method*)
SetDirectory[
 NotebookDirectory[]]; (*Set the directory for exporting the current c\
location of this file*)
exp = Export["a.dxf", 
  a]; 
(***********************************)

(*Cross[D[#1,Phi],D[#1,Psi]]&@@func *) (*Calculation of the area through \
intgeral formula*)

(***************************************************)
(*Steps*)
d1 = D[func1, Phi];(*Derivative Phi*)
d2 = D[func1, Psi];(*Derivative Psi*)
cr = Cross[d1, d2];(*Derivative intersection vector d1 и d2*)
(*Formula
Cross[{a,b,c},{x,y,z}]
{-c y+b z,c x-a z,-b x+a y}
*)
(* 1024=32*32
912=28,5*32 *)
nr = Norm[cr]; (*Vector length cr*)
nrt = TraditionalForm[
  nr] ;(*Derivation of the sub-expression in the traditional form*)
int1 = Hold[
  Integrate[\[Sqrt](Abs[-1024 cos(Psi) cos^2(Phi) - 
912.` Psi cos(Psi) cos(Phi) + 1024 cos(Psi) cos(Phi) + 
0.`]^2 + Abs[
  912.` Psi sin(Phi) cos^2(Psi) + 
1024 cos(Phi) sin(Phi) cos^2(Psi) - 
1024 sin(Phi) cos^2(Psi) + 912.` Psi sin(Phi) sin^2(Psi) + 
1024 cos(Phi) sin(Phi) sin^2(Psi) - 
1024 sin(Phi) sin^2(Psi) + 0.`]^2 + Abs[
  1024 sin(Psi) cos^2(Phi) + 912.` Psi sin(Psi) cos(Phi) - 
1024 sin(Psi) cos(Phi) + 0.`]^2), {Phi, 0, PhiEnd}, {Psi, 0, 
PsiEnd}]];
(*The withdrawal of integral in traditional form*)
int2 = Integrate[nr, {Phi, 0, PhiEnd}, {Psi, 0, PsiEnd}];
inthold = TraditionalForm[int1];
(***************************************************************)
(*One expression*)
abs = Evaluate[
  Norm[Cross[D[#1, Phi], D[#1, Psi]] & @@ 
func]];(*Integral area formula*)
int = Integrate[abs, {Phi, 0, PhiEnd}, {Psi, 0, 
   PsiEnd}]; (*Area calculation using the integral formula*)
   (*Timing[Integrate[Evaluate@Norm@Cross[D[#1,Phi],D[#1,Psi]],##2]&@@\func]*)
conv = int/ ar ;

The surface area was calculated..how to calculate surface volume ...? enter image description here

by coordinate equations, the surface was obtained by integration, and a mathematical expression was obtained

Please help me to get the expressions of the volume of the figure, similar to the calculation of the surface area, according to the specified conditions

Wolfram file https://drive.google.com/open?id=1SOjBe3l_riCWo_Ad2LiQc2Og8ZA0dwLq

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  • $\begingroup$ Ah, this looks better now. $\endgroup$ – Henrik Schumacher Apr 11 '19 at 15:18
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    $\begingroup$ These many sin(Phi), cos^2(Psi)m etc. might cause you some trouble as this is not correct Mathematica syntax. Rember: Built-in symbols always start with a capital letter and function arguments ought to be embraced by brackets ([ ]), not by parentheses (( )). $\endgroup$ – Henrik Schumacher Apr 11 '19 at 15:20
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    $\begingroup$ @Alexsey S Your question is not clear how to calculate surface volume?. The surface has no volume. Apparently there is a question about the volume of the 3D region bounded by the surface. But from your data it is not clear how the 3D region is built. Try to use R=DiscretizeRegion[ ParametricRegion[{x, y, z} /. rb1 -> r, {{Phi, 0, PhiEnd}, {Psi, 0, PsiEnd}, {r, 28.5, 33.5}}]] and RegionMeasure[R]. (Out[]=115.637) $\endgroup$ – Alex Trounev Apr 12 '19 at 12:20
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    $\begingroup$ Cross-posted community.wolfram.com/groups/-/m/t/1658725 $\endgroup$ – Michael E2 Apr 13 '19 at 3:33

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