In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... Manufacturing (from Latin manu factura, making by hand) is the use of tools and labor to make things for use or sale. ... In geometry, two sets of points are of the same shape precisely if one can be transformed to another by dilating (i. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ...

Assuming that the figure lies entirely on one side of the axis, the solid's volume is equal to the length of the circle described by the figure's centroid, times the figure's area. For other uses, see Volume (disambiguation). ... For other uses of this word, see Length (disambiguation). ... Circle illustration This article is about the shape and mathematical concept of circle. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... This article is about the physical quantity. ...

Rotating a curve

A representative disk is three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length "w") around some axis (located "r" units away); such that, a cylindrical volume, of π∫r²w units, is enclosed. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... 2-dimensional renderings (ie. ... In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. ... This article is about rotation as a movement of a physical body. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... For other uses of this word, see Length (disambiguation). ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... For other uses, see Volume (disambiguation). ...

Methods of finding volume: disc and shell methods

With these methods, it is easiest to draw the graph(s) in question, identify the area that is actually being revolved about the axis of revolution, and then draw a straight line, vertical for functions defined in terms of x and horizontal for functions defined in terms of y, which is referred to as a slice. Note that although all formulas are listed in terms of x, the formulas are exactly the same for functions defined in terms of y (with rotations about the x- and y-axes appropriately reversed).

Disc method

This is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when you are integrating along the axis of revolution. In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ...

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the x-axis is given by

If g(x) = 0 (e.g. revolving an area between curve and x-axis), this reduces to:

To visualize how this works, consider a thin vertical rectangle at x between y = f(x) on top and y = g(x) on the bottom, and revolve it about the x-axis; it forms a ring (or disc in the case that g(x) = 0), with outer radius f(x) and inner radius g(x). The area of a ring is π(R² − r²), where R is the outer radius (in this case f(x)), and r is the inner radius (in this case g(x)). Summing up all of the areas along the interval gives you the total volume.

Shell method

This is used when the slice that was drawn is parallel to the axis of revolution; i.e. when you are integrating perpendicular to the axis of revolution. Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ...

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the y-axis is given by

If g(x) = 0 (e.g. revolving an area between curve and x-axis), this reduces to:

To visualize how this works, consider a thin vertical rectangle at x with height [f(x) − g(x)], and revolve it about the y-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is 2πrh, where r is the radius (in this case x), and h is the height (in this case [f(x) − g(x)]). Summing up all of the surface areas along the interval gives you the total volume.

See also

• surface of revolution
• Gabriel's Horn
• Guldinus theorem

The parabola y=x2 rotated about the z-axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. ... Gabriels Horn (also called Torricellis trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. ... Pappuss centroid theorem consists of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. ...

External links

• Solid of Revolution at MathWorld
• Plot a solid of revolution