Semicircle

Author

Eric W. Weisstein
February 20, 2012

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Semicircle.nb.

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

©2012 Wolfram Research, Inc. except for portions noted otherwise

Curve

curve=a {Cos[t],Sin[t]}

ParametricPlot[{Sqrt[1-x^2],x},{x,-1,1},Ticks->None,ImageSize->Small]

Arc Length

x[y_]:=Sqrt[r^2-y^2]

Integrate[Sqrt[1+x'[y]^2],{y,-r,r},Assumptions->r>0]

Centroid

lx=Integrate[r,{y,-r,r}]

lx/l

ly=Integrate[y Sqrt[1+x'[y]^2],{y,-r,r},Assumptions->r>0]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&0<t<π]

Lamina

$TextStyle={FontFamily->"Times",FontSlant->"Italic",FontSize->12};

V5.2

<<Graphics`InequalityGraphics`

InequalityPlot[x^2+y^2<1&&x>=0,{x,-1,1},{y,-1,1},Ticks->False,Epilog->{
Line[{{0,-1},{0,1}}],PointSize[.07],
Point[{2/Pi,0}],Text[OverBar[(Subscript[x, 1])],{2/Pi,.12}],
Point[{4/(3Pi),0}],Text[OverBar[(Subscript[x, 2])],{4/(3Pi),.12}]
}]

V6

RegionPlot[x^2+y^2<1&&x>=0,{x,-1,1},{y,-1,1},ImageSize->Small]

Area

a=Integrate[1,{x,0,r},{y,-Sqrt[r^2-x^2],Sqrt[r^2-x^2]},Assumptions->r>0]

Integrate[Boole[x^2+y^2<r^2&&x>0],{x,-∞,∞},{y,-∞,∞}]

Integrate[Boole[x^2+y^2<r^2&&x>0],{x,-∞,∞},{y,-∞,∞},Assumptions->r>0]

Area Moment of Inertia

About Axis

Integrate[{{(y^2),x y},{x y,(x^2)}}Boole[x^2+y^2<a^2],{x,-a,a},{y,0,a},Assumptions->a>0]

N[%]

About Centroid

Integrate[{{(y^2),x y},{x y,(x^2)}}Boole[x^2+(y+ybar)^2<a^2],{x,-a,a},{y,-ybar,a-ybar},Assumptions->a>0]

N[%]

π/8-8/(9π)//N

Centroid

ax=Integrate[x,{x,0,r},{y,-Sqrt[r^2-x^2],Sqrt[r^2-x^2]},Assumptions->r>0]

ax/a

ybar=((4 a)/(3 π));

Slope

Slope[a {Cos[t],Sin[t]},t]//FullSimplify

Slope[a,q]//FullSimplify

Slope[x^2+y^2==a^2,{x,y}]//FullSimplify

Semicircular Wedge

With[{ρ=(M/((1/2)π (a^2)h))},ρ Integrate[InertiaTensor[{x,y,z}]Boole[x^2+y^2<a^2],{x,0,a},{y,-a,a},{z,-h/2,h/2},Assumptions->a>0&&h>0]]

With[{ρ=(M/((1/2)π (a^2)h))},ρ Integrate[r InertiaTensor[{r,θ,z},"CoordinateSystem"->"Cylindrical"],{r,0,a},{θ,-π/2,π/2},{z,-h/2,h/2},Assumptions->a>0&&h>0]]