In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a solid of revolution is a
solid figure
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
obtained by rotating a plane figure around some
straight line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
(the ''
axis of revolution
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is th ...
'') that lies on the same plane. The
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 ...
created by this revolution and which bounds the solid is the
surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
.
Assuming that the curve does not cross the axis, the solid's
volume
Volume is a measure of occupied 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). The de ...
is equal to the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
described by the figure's
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
multiplied by the figure's
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
(
Pappus's second centroid theorem).
A representative disc is a three-
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
of a solid of revolution. The element is created by
rotating
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
a
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
(of
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
) around some axis (located units away), so that a
cylindrical volume
Volume is a measure of occupied 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). The de ...
of units is enclosed.
Finding the volume
Two common methods for finding the volume of a solid of revolution are the
disc method and the
shell method of integration. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness , or a cylindrical shell of width ; and then find the limiting sum of these volumes as approaches 0, a value which may be found by evaluating a suitable integral. A more rigorous justification can be given by attempting to evaluate a
triple integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-numbe ...
in
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
with two different orders of integration.
Disc method
The disc method is used when the slice that was drawn is ''perpendicular to'' the axis of revolution; i.e. when integrating ''parallel to'' the axis of revolution.
The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by
:
If (e.g. revolving an area between the curve and the -axis), this reduces to:
:
The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius . The area of a ring is , where is the outer radius (in this case ), and is the inner radius (in this case ). The volume of each infinitesimal disc is therefore . The limit of the Riemann sum of the volumes of the discs between and becomes integral (1).
Assuming the applicability of
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D):
:
Cylinder method
The cylinder method is used when the slice that was drawn is ''parallel to'' the axis of revolution; i.e. when integrating ''perpendicular to'' the axis of revolution.
The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by
:
If (e.g. revolving an area between curve and -axis), this reduces to:
:
The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell. The lateral surface area of a cylinder is , where is the radius (in this case ), and is the height (in this case ). Summing up all of the surface areas along the interval gives the total volume.
This method may be derived with the same triple integral, this time with a different order of integration:
:
Parametric form
When a curve is defined by its
parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by
:
:
Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given by
:
:
Polar form
For a polar curve
where
, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are
:
:
The areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given
:
:
:
See also
*
Gabriel's Horn
Gabriel's horn (also called Torricelli's trumpet) is a particular geometry, geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Last ...
*
Guldinus theorem
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.
Th ...
*
Pseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface ...
*
Surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
*
Ungula
In solid geometry, an ungula is a region of a solid of revolution, cut off by a plane oblique to its base. A common instance is the spherical wedge. The term ''ungula'' refers to the hoof of a horse, an anatomical feature that defines a class of ...
Notes
References
*
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{{Authority control
Integral calculus
Solids