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In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s dealing with the
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
s and
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
s of
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s and
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
s of revolution. The theorems are attributed to
Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.


The first theorem

The first theorem states that the
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
''A'' of a
surface of revolution A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersec ...
generated by rotating a
plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
''C'' about an
axis An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
external to ''C'' and on the same plane is equal to the product of the
arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
''s'' of ''C'' and the distance ''d'' traveled by the geometric centroid of ''C'': A = sd. For example, the surface area of the
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
with minor
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
''r'' and major radius ''R'' is A = (2\pi r)(2\pi R) = 4\pi^2 R r.


Proof

A curve given by the positive function f(x) is bounded by two points given by: a \geq 0 and b \geq a If dL is an infinitesimal line element tangent to the curve, the length of the curve is given by: L = \int_a^b dL = \int_a^b \sqrt = \int_a^b \sqrt \, dx The y component of the centroid of this curve is: \bar = \frac \int_a^b y \, dL = \frac \int_a^b y \sqrt \, dx The area of the surface generated by rotating the curve around the x-axis is given by: A = 2 \pi \int_a^b y \, dL = 2 \pi \int_a^b y \sqrt \, dx Using the last two equations to eliminate the integral we have: A = 2 \pi \bar L


The second theorem

The second theorem states that the
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
''V'' of a
solid of revolution In geometry, a solid of revolution is a Solid geometry, solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution''), which may not Intersection (geometry), intersect the generatrix (except at its bound ...
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''. (The centroid of ''F'' is usually different from the centroid of its boundary curve ''C''.) That is: V = Ad. For example, the volume of the
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
with minor radius ''r'' and major radius ''R'' is V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2. This special case was derived by
Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
using infinitesimals.


Proof 1

The area bounded by the two functions: y = f(x) , \, \qquad y \geq 0 y = g(x) , \, \qquad f(x) \geq g(x) and bounded by the two lines: x = a \geq 0 and x = b \geq a is given by: A = \int_a^b dA = \int_a^b (x) - g(x)\, dx The x component of the centroid of this area is given by: \bar = \frac \, \int_a^b x \, (x) - g(x)\, dx If this area is rotated about the y-axis, the volume generated can be calculated using the shell method. It is given by: V = 2 \pi \int_a^b x \, (x) - g(x)\, dx Using the last two equations to eliminate the integral we have: V = 2 \pi \bar A


Proof 2

Let A be the area of F, W the solid of revolution of F, and V the volume of W. Suppose F starts in the xz-plane and rotates around the z-axis. The distance of the centroid of F from the z-axis is its x-coordinate R = \frac, and the theorem states that V = Ad = A \cdot 2\pi R = 2\pi\int_F x\,dA. To show this, let F be in the ''xz''-plane, parametrized by \mathbf(u,v) = (x(u,v),0,z(u,v)) for (u,v)\in F^*, a parameter region. Since \boldsymbol is essentially a mapping from \mathbb^2 to \mathbb^2, the area of F is given by the
change of variables In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become si ...
formula: A = \int_F dA = \iint_ \left, \frac\\,du\,dv = \iint_ \left, \frac \frac - \frac \frac\\,du\,dv, where \left, \tfrac\ is the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
of the change of variables. The solid W has the toroidal parametrization \boldsymbol(u,v,\theta) = (x(u,v)\cos\theta,x(u,v)\sin\theta,z(u,v)) for (u,v,\theta) in the parameter region W^* = F^*\times ,2\pi/math>; and its volume is V = \int_W dV = \iiint_ \left, \frac\\,du\,dv\,d\theta. Expanding, \begin \left, \frac\ & = \left, \det\begin \frac\cos\theta & \frac\cos\theta & -x\sin\theta \\ pt\frac\sin\theta & \frac\sin\theta & x\cos\theta \\ pt\frac & \frac & 0 \end\ \\ pt& = \left, -\frac\frac\,x + \frac\frac\,x\ =\ \left, -x\,\frac\ = x\left, \frac\. \end The last equality holds because the axis of rotation must be external to F, meaning x \geq 0. Now, \begin V &= \iiint_ \left, \frac\\,du\,dv\,d\theta \\ ex&= \int_0^\!\!\!\!\iint_ x(u,v)\left, \frac\ du\,dv\,d\theta \\ pt& = 2\pi\iint_ x(u,v)\left, \frac\\,du\,dv \\ ex&= 2\pi\int_F x\,dA \end by change of variables.


Generalizations

The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions. Goodman & Goodman generalize the second theorem as follows. If the figure moves through space so that it remains
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to the curve traced by the centroid of , then it sweeps out a solid of volume , where is the area of and is the length of . (This assumes the solid does not intersect itself.) In particular, may rotate about its centroid during the motion. However, the corresponding generalization of the first theorem is only true if the curve traced by the centroid lies in a plane perpendicular to the plane of .


In ''n''-dimensions

In general, one can generate an n dimensional solid by rotating an n-p dimensional solid F around a p dimensional sphere. This is called an n-solid of revolution of species p. Let the p-th centroid of F be defined by R = \frac, Then Pappus' theorems generalize to:
Volume of n-solid of revolution of species p
= (Volume of generating (np)-solid) \times (Surface area of p-sphere traced by the p-th centroid of the generating solid)
and
Surface area of n-solid of revolution of species p
= (Surface area of generating (np)-solid) \times (Surface area of p-sphere traced by the p-th centroid of the generating solid)
The original theorems are the case with n=3,\, p = 1.


Footnotes


References


External links

*{{MathWorld, title=Pappus's Centroid Theorem, urlname=PappussCentroidTheorem Theorems in calculus Geometric centers Theorems in geometry Area Volume Greek mathematics