Brstjjy

The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation.

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of Alexandria^[a] and Paul Guldin.^[b]

The first theorem

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C:

A=sd.{displaystyle A=sd.}

For example, the surface area of the torus with minor radius r and major radius R is

A=(2πr)(2πR)=4π2Rr.{displaystyle A=(2pi r)(2pi R)=4pi ^{2}Rr.}

The second theorem

The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (Note that the centroid of F is usually different from the centroid of its boundary curve C.) That is:

V=Ad.{displaystyle V=Ad.}

For example, the volume of the torus with minor radius r and major radius R is

V=(πr2)(2πR)=2π2Rr2.{displaystyle V=(pi r^{2})(2pi R)=2pi ^{2}Rr^{2}.}

This special case was derived by Johannes Kepler using infinitesimals.^[c]

Proof

Let A{displaystyle A} be the area of F{displaystyle F}, W{displaystyle W} the solid of revolution of F{displaystyle F}, and V{displaystyle V} the volume of W{displaystyle W}. Suppose F{displaystyle F} starts in the xz{displaystyle xz}-plane and rotates around the z{displaystyle z}-axis. The distance of the centroid of F{displaystyle F} from the z{displaystyle z}-axis is its x{displaystyle x}-coordinate

R=∬FxdAA,{displaystyle R={frac {iint _{F}x,dA}{A}},}

and the theorem states that

V=Ad=A⋅2πR=2π∬FxdA.{displaystyle V=Ad=Acdot 2pi R=2pi iint _{F}x,dA.}

To show this, let F{displaystyle F} be in the xz-plane, parametrized by Φ(u,v)=(x(u,v),0,z(u,v)){displaystyle mathbf {Phi } (u,v)=(x(u,v),0,z(u,v))} for (u,v)∈F∗{displaystyle (u,v)in F^{*}}, a parameter region. Since Φ{displaystyle mathbf {Phi } } is essentially a mapping from R2{displaystyle mathbb {R} ^{2}} to R2{displaystyle mathbb {R} ^{2}}, the area of F{displaystyle F} is given by the change of variables formula:

A=∬FdA=∬F∗|∂(x,z)∂(u,v)|dudv=∬F∗|∂x∂u∂z∂v−∂x∂v∂z∂u|dudv,{displaystyle A=iint _{F}dA=iint _{F^{*}}left|{frac {partial (x,z)}{partial (u,v)}}right|,du,dv=iint _{F^{*}}left|{frac {partial x}{partial u}}{frac {partial z}{partial v}}-{frac {partial x}{partial v}}{frac {partial z}{partial u}}right|,du,dv,}

where ∂(x,z)∂(u,v){displaystyle {tfrac {partial (x,z)}{partial (u,v)}}} is the determinant of the Jacobian matrix of the change of variables.

The solid W{displaystyle W} has the toroidal parametrization Φ(u,v,θ)=(x(u,v)cos⁡θ,x(u,v)sin⁡θ,z(u,v)){displaystyle mathbf {Phi } (u,v,theta )=(x(u,v)cos theta ,x(u,v)sin theta ,z(u,v))} for (u,v,θ){displaystyle (u,v,theta )} in the parameter region W∗=F∗×[0,2π]{displaystyle W^{*}=F^{*}times [0,2pi ]}; and its volume is

V=∭WdV=∭W∗|∂(x,y,z)∂(u,v,θ)|dudvdθ.{displaystyle V=iiint _{W}dV=iiint _{W^{*}}left|{frac {partial (x,y,z)}{partial (u,v,theta )}}right|,du,dv,dtheta .}

Expanding,

|∂(x,y,z)∂(u,v,θ)|=|det[∂x∂ucos⁡θ∂x∂vcos⁡θ−xsin⁡θ∂x∂usin⁡θ∂x∂vsin⁡θxcos⁡θ∂z∂u∂z∂v0]|=|−∂z∂v∂x∂ux+∂z∂u∂x∂vx|= |−x∂(x,z)∂(u,v)|=x|∂(x,z)∂(u,v)|.{displaystyle {begin{aligned}left|{frac {partial (x,y,z)}{partial (u,v,theta )}}right|&=left|det {begin{bmatrix}{frac {partial x}{partial u}}cos theta &{frac {partial x}{partial v}}cos theta &-xsin theta \[6pt]{frac {partial x}{partial u}}sin theta &{frac {partial x}{partial v}}sin theta &xcos theta \[6pt]{frac {partial z}{partial u}}&{frac {partial z}{partial v}}&0end{bmatrix}}right|\[5pt]&=left|-{frac {partial z}{partial v}}{frac {partial x}{partial u}},x+{frac {partial z}{partial u}}{frac {partial x}{partial v}},xright|= left|-x,{frac {partial (x,z)}{partial (u,v)}}right|=xleft|{frac {partial (x,z)}{partial (u,v)}}right|.end{aligned}}}

The last equality holds because the axis of rotation must be external to F{displaystyle F}, meaning x≥0{displaystyle xgeq 0}. Now,

V=∭W∗|∂(x,y,z)∂(u,v,θ)|dudvdθ=∫02π∬F∗x(u,v)|∂(x,z)∂(u,v)|dudvdθ=2π∬F∗x(u,v)|∂(x,z)∂(u,v)|dudv=2π∬FxdA{displaystyle {begin{aligned}V&=iiint _{W^{*}}left|{frac {partial (x,y,z)}{partial (u,v,theta )}}right|,du,dv,dtheta =int _{0}^{2pi }!!!!iint _{F^{*}}x(u,v)left|{frac {partial (x,z)}{partial (u,v)}}right|,du,dv,dtheta \[6pt]&=2pi iint _{F^{*}}x(u,v)left|{frac {partial (x,z)}{partial (u,v)}}right|,du,dv=2pi iint _{F}x,dAend{aligned}}}

by change of variables.

Generalizations

The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions.

Goodman & Goodman^[5] generalize the second theorem as follows. If the figure F moves through space so that it remains perpendicular to the curve L traced by the centroid of F, then it sweeps out a solid of volume V = Ad, where A is the area of F and d is the length of L. (This assumes the solid does not intersect itself.) In particular, F may rotate about its centroid during the motion.

However, the corresponding generalization of the first theorem is only true if the curve L traced by the centroid lies in a plane perpendicular to the plane of C.

Footnotes

References

External links

Wikimedia Commons has media related to Pappus-Guldinus theorem.

• Weisstein, Eric W. "Pappus's Centroid Theorem". MathWorld.