Cylindrical Segment

Author

Eric W. Weisstein
November 18, 2010

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

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Cut by Oblique Plane

Figure

<<ExtraPackages`Hidekazu`Knife`

cylseg=Show[
knife[cylinder[1,2],{1,1.5,3,-3},1],Boxed->False]

Volume

By Planes

Integrate[Sqrt[R^2-x^2]((x/R)((Subscript[h, 2])-(Subscript[h, 1]))+((Subscript[h, 1])+(Subscript[h, 2]))),{x,-R,R},Assumptions->R>0]

By 2-D Integration

h[r_,t_]:=(Subscript[h, 1])+1/2(1+Cos[t]r/R)((Subscript[h, 2])-(Subscript[h, 1]))

Simplify[Integrate[h[r,t]r,{r,0,R},{t,0,2Pi}]]

Simplify[Integrate[r,{r,0,R},{t,0,2Pi},{z,0,h[r,t]}]]

Simplify[Cos[ArcTan[y/x]],(x|y)∈Reals]

h[r,t] /. {Cos[t] ->x/Sqrt[x^2+y^2],r->Sqrt[x^2+y^2]}

vol=Simplify[Integrate[1,{x,-R,R},{y,-Sqrt[R^2-x^2],Sqrt[R^2-x^2]},{z,0,(Subscript[h, 1])+(1/2) (1+(x/R)) (-(Subscript[h, 1])+(Subscript[h, 2]))}]/.Sign[R]->1]

Lateral Surface Area

Integrate[R h[R,t],{t,0,2π}]//Simplify

Centroids

x-Centroid

volx=Simplify[Integrate[x,{x,-R,R},{y,-Sqrt[R^2-x^2],Sqrt[R^2-x^2]},{z,0,(Subscript[h, 1])+(1/2) (1+(x/R)) (-(Subscript[h, 1])+(Subscript[h, 2]))}]/.Sign[R]->1]

volx/vol

% /. (Subscript[h, 2])->(Subscript[h, 1])

y-Centroid

voly=Simplify[Integrate[y,{x,-R,R},{y,-Sqrt[R^2-x^2],Sqrt[R^2-x^2]},{z,0,(Subscript[h, 1])+(1/2) (1+(x/R)) (-(Subscript[h, 1])+(Subscript[h, 2]))}]/.Sign[R]->1]

voly/vol

z-Centroid

Simplify[Integrate[z,{x,-R,R},{y,-Sqrt[R^2-x^2],Sqrt[R^2-x^2]},{z,0,(Subscript[h, 1])+(1/2) (1+(x/R)) (-(Subscript[h, 1])+(Subscript[h, 2]))}]/.Sign[R]->1]

Simplify[Integrate[z r,{r,0,R},{t,0,2Pi},{z,0,((Subscript[h, 1])+h[r,t])}]]

Simplify[Integrate[z r,{r,0,R},{t,0,2Pi},{z,0,((Subscript[h, 1])+h[r,t])}]/Integrate[r,{r,0,R},{t,0,2Pi},{z,0,((Subscript[h, 1])+h[r,t])}]]

% /. (Subscript[h, 2])->(Subscript[h, 1])