Cylindrical Wedge

Author

Eric W. Weisstein
August 14, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/CylindricalWedge.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CylindricalWedge.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=z≥0&&z+(h-z) Cos[ϕ]≤h x/R&&x^2+y^2≤R^2;

implreg=ImplicitRegion[ineq,{x,y,z}];

reg=RegionIntersection[ImplicitRegion[z≥0&&z+(h-z) Cos[ϕ]≤h x/R,{x,y,z}],Cylinder[{{0,0,0},{0,0,h}},R]];

assum=h>0&&R>0&&0<ϕ<π/2;

Plots

Graphics3D

V5

<<Utilities`LiveGraphics3D`

CylindricalWedge[a_,b_,h_,n_:5]:=Module[{ang=ArcTan[a-b,Sqrt[a^2-(a-b)^2]],t,bot,top},
bot=Table[{Cos[#],Sin[#],0}&[t],{t,-ang,ang,ang/(2n)}];
top=Table[{Cos[#],Sin[#],h(Cos[t]-(a-b))/b}&[t],{t,-ang,ang,ang/(2n)}];
Append[Polygon/@{Reverse[bot],top},Polygon/@(Flatten[#,1]&/@Transpose[{Partition[bot,2,1],Reverse/@Partition[top,2,1]}])]
]

WriteLiveForm["cylwedge.m",Show[Graphics3D[CylindricalWedge[1,1.5,1,5]],Boxed->False,
ViewPoint->{-.5,-1,.2}]]//Timing

V6 Graphics3D

Graphics3D[CylindricalWedge[1,1.5,1,5],Boxed->False]

Entity Image

Show[SolidData["CylindricalWedge","Image"],ImageSize->Small,Method->{"ShrinkWrap"->True}]

Entity ImplicitRegion

Show[DiscretizeRegion@SolidData["CylindricalWedge","ImplicitRegion"][2,1,1],ImageSize->Small]

Entity ParametricRegion

Show[DiscretizeRegion@SolidData["CylindricalWedge","ParametricRegion"][2,1,1],ImageSize->Small]

Entity Region

Show[DiscretizeRegion@SolidData["CylindricalWedge","Region"][2,1,1],ImageSize->Small]

RegionPlot

Block[{h=.5,b=1,R=1.5},RegionPlot3D[ineq,{x,R-b,R},{y,-(Sqrt[b(2R-b)]),(Sqrt[b(2R-b)])},{z,0,h},BoxRatios->Automatic,MaxRecursion->5,PlotPoints->50]]

Polygon Stack

With[{h=1.5,b=1.5,R=1},
Show[Graphics3D[Table[
y=Sqrt[R^2-x^2];z=h(x-R+b)/b;
Polygon[{{x,-y,0},{x,y,0},{x,y,z},{x,-y,z}}],{x,R-b,R,b/35}]],
Boxed->False,
ViewPoint->2{-.5,-1,.2},
LightSources->{{{1.`,0.`,1.`},RGBColor[1,0,0]},{{1.`,1.`,1.`},RGBColor[0,1,0]},{{0.`,1.`,1.`},RGBColor[0,0,1]},{{-1,0,1},RGBColor[.8,.9,1]}}
]
]

DiscretizeRegion

With[{R=2,h=1,ϕ=1},Show[DiscretizeRegion[reg],ImageSize->Small,Method->{"ShrinkWrap"->True}]]

Properties

Equation

Det[{{x,y,z,1},
{R-b,a,0,1},
{R-b,-a,0,1},
{R,0,h,1}}]//HoldForm//TraditionalForm

Collect[Simplify[Det[{{x,y,z,1},
{R-b,a,0,1},
{R-b,-a,0,1},
{R,0,h,1}}]/(2a)],h]

Simplify[Solve[-b h+h R-h x+b z==0,z]]

Simplify[R^2-(b-R)^2]

ineqs=z>0&&h(R-b-x)+b z<0&&x^2+y^2<R^2

z>0&&h(R-b-x)+b z<0&&x^2+y^2<R^2/.{b->R(1-Cos[ϕ])}//Simplify

Centroid

ineq=Entity["Solid","CylindricalWedge"]["Inequalities"][R,h,p][x,y,z]

Assuming[{h>0,R>0,-1<Cos[p]<1,0<p<π},FullSimplify@((Integrate[{x,y,z}Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]

FullSimplify[%162/.(ArcTan[Cot[p_]]|ArcCot[Tan[p_]]):>π/2-p]

%186//TraditionalForm

FullSimplify[%-{-R (3 p-5 Cos[p] Sin[p]+2 Cos[p]^3 Sin[p])/(4 (3 p Cos[p]-2 Sin[p]-Cos[p]^2 Sin[p])),0,h (3 p+12 p Cos[p]^2-13 Cos[p] Sin[p]-2 Cos[p]^3 Sin[p])/(8 (Cos[p]-1) (3 p Cos[p]-2 Sin[p]-Cos[p]^2 Sin[p]))}]

{-((R (3 p+2 sin(p) (cos^3)(p)-5 sin(p) cos(p)))/(4 (-2 sin(p)+3 p cos(p)-sin(p) (cos^2)(p)))),0,((h (3 p+12 p (cos^2)(p)-2 sin(p) (cos^3)(p)-13 sin(p) cos(p)))/(8 (cos(p)-1) (-2 sin(p)+3 p cos(p)-sin(p) (cos^2)(p))))}

{Plot[ArcTan[Cot[p]],{p,0,π}],Plot[ArcCot[Tan[p]],{p,0,π}]}

{Plot[(π/2-p)-ArcTan[Cot[p]],{p,0,π}],Plot[(π/2-p)-ArcCot[Tan[p]],{p,0,π}]}

(xbar=Assuming[0<b<2R,FullSimplify[(1/(V[h,R,b]))Integrate[{x,y,z},{x,R-b,R},{y,-Sqrt[R^2-x^2],Sqrt[R^2-x^2]},{z,0,((h (-R+b+x))/b)}]]])//Timing

b==R check

Simplify[xbar/.b->R,R>0]

In terms of ϕ

xbar2=Factor[xbar/.{ArcTan[((b-R)/(Sqrt[-b (b-2 R)]))]->ϕ-π/2,(Sqrt[(b^(n_.)) (-b+2 R)])->R Sin[ϕ](((R(1-Cos[ϕ])))^((n-1)/2)), (Sqrt[-b (b-2 R)])->R Sin[ϕ]}/.b->R(1-Cos[ϕ])]

xbar2/.ϕ->π/2

xbar3=FullSimplify[xbar2]

xbar3/.ϕ->π/2

xbar4=Divide@@Factor[Expand[Through[{Numerator,Denominator}[xbar2]]]/.((Cos[ϕ])^(n_)):>(((1-((Sin[ϕ])^2)))^(Floor[n/2]))((Cos[ϕ])^(Mod[n,2]))]

xbar4/.ϕ->π/2

FullSimplify[xbar4]

xbar5=Divide@@Factor[Expand[Through[{Numerator,Denominator}[xbar2]]]/.((Sin[ϕ])^(n_)):>(((1-((Cos[ϕ])^2)))^(Floor[n/2]))((Sin[ϕ])^(Mod[n,2]))]

xbar5/.ϕ->π/2

FullSimplify[xbar5]

LateralSurfaceArea

s=R FullSimplify[2Integrate[((h (b-R+R Cos[t]))/b),{t,0,Pi/2+ArcTan[(b-R)/Sqrt[2b R-b^2]]}],b>R>0&&2b R>b^2]

s2=s/.ArcTan[((b-R)/(Sqrt[-b (b-2 R)]))]:>ϕ-Pi/2/.(Sqrt[-b (b-2 R)])->a

FullSimplify[s3=s2/.a->R Sin[ϕ]/.b->R(1-Cos[ϕ])]

Regions

SolidData["CylindricalWedge","ImplicitRegion"][R,h,ϕ]

SolidData["CylindricalWedge","ParametricRegion"]

SolidData["CylindricalWedge","Region"][R,h,ϕ]

Volume

Assuming[assum,FullSimplify[Integrate[1,{x,y,z}∈SolidData["CylindricalWedge","Region"][R,h,ϕ]]]]//Timing

FullSimplify[-((h (R^2) (-3 ArcCos[Cos[ϕ]] Cos[ϕ]+(2+((Cos[ϕ])^2)) (Sqrt[((Sin[ϕ])^2)])))/(3 (-1+Cos[ϕ]))),0<ϕ<π/2]

Refine[-((h (R^2) (-3 ArcCos[Cos[ϕ]] Cos[ϕ]+(2+((Cos[ϕ])^2)) (Sqrt[((Sin[ϕ])^2)])))/(3 (-1+Cos[ϕ]))),0<ϕ<π/2]

-((h (R^2) (-3 ϕ Cos[ϕ]+(2+((Cos[ϕ])^2)) Sin[ϕ]))/(3 (-1+Cos[ϕ])))/.Cos[x_]^2:>1-Sin[x]^2//Factor

Assuming[assum,FullSimplify[Integrate[1,{x,y,z}∈SolidData["CylindricalWedge","ImplicitRegion"][R,h,ϕ]]]]//Timing

RegionMeasure

Assuming[assum,FullSimplify[RegionMeasure[SolidData["CylindricalWedge","Region"][R,h,ϕ]]]]//Timing

Assuming[assum,FullSimplify[RegionMeasure[SolidData["CylindricalWedge","ImplicitRegion"][R,h,ϕ]]]]//Timing

Volume

Assuming[assum,FullSimplify[Volume[SolidData["CylindricalWedge","Region"][R,h,ϕ]]]]//Timing

Refine[%60,0<ϕ<π/2]

Integrate

v=FullSimplify[2Integrate[h(x-R+b)/b Sqrt[R^2-x^2],{x,R-b,R}],b>R>0]

(h/(6*b))*(2*Sqrt[(2R-b)*b]*(3R^2-2R b+b^2)-3π R^2*(R-b)+6R^2*(R-b)*ArcSin[(R-b)/R])

(eqn={
(h/(6b))(2  (Sqrt[-b (b-2 R)]) ((b^2)-2 b R+3 (R^2))+3(b-R) (R^2) π+6 h (b-R) (R^2)  ArcTan[((b-R)/(Sqrt[-b (b-2 R)]))]),
(h/(6 b)) (2 (Sqrt[b (-b+2 R)]) ((b^2)-2 b R+3 (R^2))-3 π (R^2) (R-b)+6 (R^2) (R-b) ArcSin[((R-b)/R)])
})//TraditionalForm

eqn/.{h->1,b->2,R->3.}

v1=FullSimplify[2Integrate[((h (-R+b+x))/b),{x,R-b,R},{y,0,Sqrt[R^2-x^2]}],b>R>0]

v2=FullSimplify[v1/.ArcTan[((b-R)/(Sqrt[-b (b-2 R)]))]->ϕ-π/2/. (Sqrt[-b (b-2 R)])->a,b>R>0]

v3=FullSimplify[v2/.b^2->2b R-a^2]

v4=FullSimplify[v3/.a->R Sin[ϕ]/.b->R-R Cos[ϕ]]

vv=h R^3/3/b(2 Sin[ϕ]-Cos[ϕ](3ϕ-Sin[2ϕ]/2))

Boole

ineqs[h_,b_,R_]:=z>0&&h(R-b-x)+b z<0&&x^2+y^2<R^2

Integrate[Boole[ineqs[h,b,R]],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->h>0&&0<R<b<2R]

FullSimplify[Refine[FullSimplify[%,h>0&&0<R<b<2R],b>0],h>0&&0<R<b<2R]

Integrate[Boole[ineqs[h,R,R]],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->h>0&&0<R]

Integrate[Boole[ineqs[h,2R,R]],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->h>0&&0<R]

Symmetrical Check

Simplify[v3/.a->Sqrt[2b R-b^2]/.ϕ->Pi/2/.b->R,R>0]

Simplify[v4/.ϕ->Pi/2]

Numerical Check

Block[{h=.5,R=1.5,ϕ=1.2},{
vol=NIntegrate[Boole[R (z+(h-z) Cos[ϕ])<h x&&(x^2)+(y^2)<(R^2)],{x,R Cos[ϕ],R},{y,-R Sin[ϕ],R Sin[ϕ]},{z,0,h}],
((h (R^2) (3 ϕ Cos[ϕ]-3 Sin[ϕ]+((Sin[ϕ])^3)))/(3 (-1+Cos[ϕ])))
}]

Volume

V[h_,R_,b_]:=((2 h (Sqrt[-b (b-2 R)]) ((b^2)-2 b R+3 (R^2))+3 h (b-R) (R^2) (π+2 ArcTan[((b-R)/(Sqrt[-b (b-2 R)]))]))/(6 b))

Assuming[0<b<2R,FullSimplify[(1/(V[h,R,b]))Integrate[1,{x,R-b,R},{y,-Sqrt[R^2-x^2],Sqrt[R^2-x^2]},{z,0,((h (-R+b+x))/b)}]]]//Timing

Semicircular cylindrical wedge

Called CylidricalHoof in MW up until August 2023

Construction

Entity["Solid","SolidCylinder"]["Graphics3D"]

Entity["Solid","SolidCylinder"]["Region"]

Entity["Solid","SolidCylinder"]["Inequalities"][a,h][x,y,z]

ineq[a_,h_]:=(x^2)+(y^2)≤(a^2)&&0≤z≤h&&0≤z≤h&&x≥0&&a z≤h x

Table[RegionPlot3D[ineq[1,h],{x,.01,1},{y,-1,1},{z,-.01,h},PlotPoints->50,BoxRatios->Automatic,AxesLabel->{"x","y","z"}],{h,{1/2,1,2}}]

CrossSections

{x1,y1,z1}={0,-r,0};
{x2,y2,z2}={0,r,0};
{x3,y3,z3}={-r,0,h};
Factor[Det[{{x,y,z,1},{x1,y1,z1,1},{x2,y2,z2,1},{x3,y3,z3,1}}]]

ParametricPlot3D[Evaluate[{{x,Sqrt[r^2-x^2],h x/r},{x,-Sqrt[r^2-x^2],h x/r}}/.{h->5,r->1}],{x,0,r},Ticks->None]

Rx[t_]:={{1,0,0},{0,Cos[t],Sin[t]},{0,-Sin[t],Cos[t]}}
Rz[t_]:={{Cos[t],Sin[t],0},{-Sin[t],Cos[t],0},{0,0,1}}

Simplify[Rz[ArcTan[h/r]].(Rx[Pi/2].{x,Sqrt[r^2-x^2],h x/r})]

N[Sqrt[r^2+h^2]]

h=5;r=1;
ParametricPlot[{
   {x/r Sqrt[r^2+h^2],Sqrt[r^2-x^2]},
   {x/r Sqrt[r^2+h^2],-Sqrt[r^2-x^2]}
 },{x,0,r},
 AspectRatio->Automatic,Ticks->None]

h=.;
x[t_]:=t/r Sqrt[r^2+h^2]
y[t_]:=Sqrt[r^2-t^2]
dsdt[t_]:=Sqrt[Simplify[x'[t]^2+y'[t]^2]]

Plot[Evaluate[dsdt[t] /. {h->5,r->1}],{t,0,1}]

dsdt[t]

Integrate[dsdt[t],{t,0,r}]

Properties

Entity["Solid","CylindricalWedge"]["Inequalities"][R,h,Pi/2][x,y,z]

ineq=0<=z<=h x/R&&x^2+y^2<=R^2

Centroid

Entity["Solid","CylindricalWedge"]["Centroid"][R,h,Pi/2]

Assuming[{h>0,R>0},((Integrate[{x,y,z}Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]

FullSimplify[Integrate[1,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

xbar=FullSimplify[Integrate[x ,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

PowerExpand[Simplify[xbar /. b->a]]

xbar=FullSimplify[Integrate[x ,{x,0,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h x)/a)}]/V[h,a,a],a>0]

ybar=FullSimplify[Integrate[y,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

zbar=FullSimplify[Integrate[z ,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

PowerExpand[Simplify[zbar /. b->a]]

GeneralizedDiameter

Assuming[R>0,Simplify@MaxValue[{x^2+y^2+z^2,ineq},{x,y,z}]]

InertiaTensor

Assuming[{h>0,R>0},FullSimplify@((Integrate[InertiaMatrix[{x,y,z}]Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]

LateralSurfaceArea

Entity["Solid","CylindricalWedge"]["LateralSurfaceArea"][R,h,Pi/2]

MeanCylindricalRadius

Assuming[{h>0,R>0},FullSimplify@((Integrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]

MeanSphericalRadius

Block[{R=1,h=1},((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/((2 h (R^2))/3))]

Assuming[{h>0,R>0},Simplify@((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]//Timing

FullSimplify[(-16 π (R^3) (x^3)+8 h R x (Sqrt[((R^2)-(x^2)) ((R^4)+(h^2) (x^2))])+8 (R^3) (x^3) ArcCot[(x/(Sqrt[(R^2)-(x^2)]))]+8 (R^3) (x^3) ArcTan[(x/(Sqrt[(R^2)-(x^2)]))]-2 (h^3) (x^3) Log[R]+6 h (R^2) (x^3) Log[R]+4 (R^5) (Sqrt[(R^2)-(x^2)]) Log[R]+8 (R^3) (x^2) (Sqrt[(R^2)-(x^2)]) Log[R]-6 h (R^2) (x^3) Log[(h^2)+(R^2)]-12 h (R^2) (x^3) Log[x]+(h^3) (x^3) Log[((h^2)+(R^2)) (x^2)]+3 h (R^2) (x^3) Log[((h^2)+(R^2)) (x^2)]+3 (h^3) (x^3) Log[((((h^2)+(R^2)) (x^2))/(R^2))]+3 h (R^2) (x^3) Log[((((h^2)+(R^2)) (x^2))/(R^2))]-4 (R^5) (Sqrt[(R^2)-(x^2)]) Log[((-h x+(Sqrt[(R^4)+(h^2) (x^2)]))/R)]-8 (R^3) (x^2) (Sqrt[(R^2)-(x^2)]) Log[((-h x+(Sqrt[(R^4)+(h^2) (x^2)]))/R)]-6 (h^3) (x^3) Log[-(Sqrt[(R^2)-(x^2)])+(Sqrt[(R^2)+(((h^2) (x^2))/(R^2))])]-6 h (R^2) (x^3) Log[-(Sqrt[(R^2)-(x^2)])+(Sqrt[(R^2)+(((h^2) (x^2))/(R^2))])]-2 (h^3) (x^3) Log[(Sqrt[(R^2)-(x^2)])+(Sqrt[(R^2)+(((h^2) (x^2))/(R^2))])]+6 h (R^2) (x^3) Log[(Sqrt[(R^2)-(x^2)])+(Sqrt[(R^2)+(((h^2) (x^2))/(R^2))])]-2 I (R^3) (x^3) Log[(((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x-I (Sqrt[(R^2)-(x^2)])))/((h^2) (x^4) (x-I (Sqrt[(R^2)-(x^2)]))))]+6 I (R^3) (x^3) Log[-((4 ((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x-I (Sqrt[(R^2)-(x^2)]))))/((h^2) (x^2) (x-I (Sqrt[(R^2)-(x^2)]))))]+2 I (R^3) (x^3) Log[(((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x+I (Sqrt[(R^2)-(x^2)])))/((h^2) (x^4) (x+I (Sqrt[(R^2)-(x^2)]))))]-6 I (R^3) (x^3) Log[-((4 ((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x+I (Sqrt[(R^2)-(x^2)]))))/((h^2) (x^2) (x+I (Sqrt[(R^2)-(x^2)]))))]),h>0&&R>0]//Timing

FullSimplify[2 (-8 π (R^3) (x^3)+4 h R x (Sqrt[(R-x) (R+x) ((R^4)+(h^2) (x^2))])+4 (R^3) (x^3) ArcCot[(x/(Sqrt[(R^2)-(x^2)]))]+4 (R^3) (x^3) ArcTan[(x/(Sqrt[(R^2)-(x^2)]))]+4 (R^5) (Sqrt[(R^2)-(x^2)]) Log[R]+8 (R^3) (x^2) (Sqrt[(R^2)-(x^2)]) Log[R]-2 (R^5) (Sqrt[(R^2)-(x^2)]) Log[-h x+(Sqrt[(R^4)+(h^2) (x^2)])]-4 (R^3) (x^2) (Sqrt[(R^2)-(x^2)]) Log[-h x+(Sqrt[(R^4)+(h^2) (x^2)])]+(h^3) (x^3) (2 Log[((h^2)+(R^2)) (x^2)]-3 Log[-R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])]-Log[R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])])+3 h (R^2) (x^3) (-2 Log[x]+Log[(x^2)]-Log[-R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])]+Log[R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])])-I (R^3) (x^3) (Log[(((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x-I (Sqrt[(R^2)-(x^2)])))/((x^4) (x-I (Sqrt[(R^2)-(x^2)]))))]-3 Log[-((I ((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x-I (Sqrt[(R^2)-(x^2)]))))/((x^2) (I x+(Sqrt[(R^2)-(x^2)]))))]-Log[(((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x+I (Sqrt[(R^2)-(x^2)])))/((x^4) (x+I (Sqrt[(R^2)-(x^2)]))))]+3 Log[((I ((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x+I (Sqrt[(R^2)-(x^2)]))))/((x^2) (-I x+(Sqrt[(R^2)-(x^2)]))))])),Assumptions->h>0&&R>0]

rxyz=2 (-8 π (R^3) (x^3)+4 h R x (Sqrt[(R-x) (R+x) ((R^4)+(h^2) (x^2))])+4 (R^3) (x^3) ArcCot[(x/(Sqrt[(R^2)-(x^2)]))]+4 (R^3) (x^3) ArcTan[(x/(Sqrt[(R^2)-(x^2)]))]+4 (R^5) (Sqrt[(R^2)-(x^2)]) Log[R]+8 (R^3) (x^2) (Sqrt[(R^2)-(x^2)]) Log[R]-2 (R^5) (Sqrt[(R^2)-(x^2)]) Log[-h x+(Sqrt[(R^4)+(h^2) (x^2)])]-4 (R^3) (x^2) (Sqrt[(R^2)-(x^2)]) Log[-h x+(Sqrt[(R^4)+(h^2) (x^2)])]+(h^3) (x^3) (2 Log[((h^2)+(R^2)) (x^2)]-3 Log[-R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])]-Log[R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])])+3 h (R^2) (x^3) (-2 Log[x]+Log[(x^2)]-Log[-R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])]+Log[R (Sqrt[(R^2)-(x^2)])+(Sqrt[(R^4)+(h^2) (x^2)])])-I (R^3) (x^3) (Log[(((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x-I (Sqrt[(R^2)-(x^2)])))/((x^4) (x-I (Sqrt[(R^2)-(x^2)]))))]-3 Log[-((I ((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x-I (Sqrt[(R^2)-(x^2)]))))/((x^2) (I x+(Sqrt[(R^2)-(x^2)]))))]-Log[(((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x+I (Sqrt[(R^2)-(x^2)])))/((x^4) (x+I (Sqrt[(R^2)-(x^2)]))))]+3 Log[((I ((h^2) x+h (Sqrt[(R^4)+(h^2) (x^2)])+(R^2) (x+I (Sqrt[(R^2)-(x^2)]))))/((x^2) (-I x+(Sqrt[(R^2)-(x^2)]))))]));

rxyz/.{R->1,h->1}

i1=Assuming[{h>0,R>0},FullSimplify@Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{x,0,R}]]

Assuming[{h>0,R>0},((Integrate[i1,{y,-R,R}])/((2 h (R^2))/3))]

Block[{R=1,h=1},((NIntegrate[i1,{y,-∞,∞},{z,-∞,∞}])/((2 h (R^2))/3))]

Assuming[{h>0,R>0},FullSimplify@Integrate[%,{y,0,R}]]

MeanSquareCylindricalRadius

Assuming[{h>0,R>0},FullSimplify@((Integrate[((x^2)+(y^2))Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]

MeanSquareSphericalRadius

Assuming[{h>0,R>0},FullSimplify@((Integrate[((x^2)+(y^2)+(z^2))Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]))]

SurfaceArea

Assuming[{h>0,R>0},Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]

Volume

Entity["Solid","CylindricalWedge"]["Volume"][R,h,Pi/2]

Assuming[{h>0,R>0},Integrate[Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]