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. Image File history File links ParabolaRotation. ... Image File history File links ParabolaRotation. ... Wikisource has an original article from the 1911 Encyclopædia Britannica about: Parabola A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... 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. ...

Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle generates a toroidal surface. A right circular cylinder In mathematics, a cylinder is a quadric, i. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... A torus. ...

If the curve is described by the functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y axis, then the area A is given by the integral

,

provided that x(t) is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity Pappuss centroid theorem consists of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. ...

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity 2πx(t) is the path of (the centroid of) this small segment, as required by Pappus's theorem. The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ... For other uses, see Curve (disambiguation). ...

If the curve is described by the function y = f(x), then the integral becomes

for revolution around the x-axis, and

for revolution around the y-axis. These come from the above formula.

For example, the spherical surface with unit radius is generated by the curve x(t)=sin(t), y(t)=cos(t), when t ranges over [0,π]. Its area is therefore A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ...

.

Applications of surfaces of revolution The use of surface of revolutions is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.

See also

• Solid of revolution
• Gabriel's Horn