area of surface of revolution

A surface of revolutionMathworldPlanetmath is a 3D surface, generated when an arc is rotated fully around a straight line.

The general surface of revolution is obtained when the arc is rotated about an arbitrary axis. If one chooses Cartesian coordinatesMathworldPlanetmath, and specializes to the case of a surface of revolution generated by rotating about the x-axis a curve described by y in the intervalMathworldPlanetmathPlanetmath [a,b], its area can be calculated by the formula


Similarly, if the curve is rotated about the y-axis rather than the x-axis, one has the following formula:


The general formula is most often seen with parametric coordinates. If x(t) and y(t) describe the curve, and x(t) is always positive or zero, then the area of the general surface of revolution A in the interval [a,b] can be calulated by the formula


To obtain a specific surface of revolution, translationPlanetmathPlanetmath or rotation can be used to move an arc before revolving it around an axis. For example, the specific surface of revolution around the line y=s can be found by replacing y with y-s, moving the arc towards the x-axis so  y=s  lies on it. Now, the surface of revolution can be found using one of the formulae above.

In this specific case, replacing y with  y=s,  the area of a surface of revolution is found using the formula

Title area of surface of revolution
Canonical name AreaOfSurfaceOfRevolution
Date of creation 2014-07-24 18:36:37
Last modified on 2014-07-24 18:36:37
Owner rspuzio (6075)
Last modified by pahio (2872)
Numerical id 13
Author rspuzio (2872)
Entry type Topic
Classification msc 53A05
Classification msc 26B15
Synonym area of revolution
Synonym surface area of revolution
Related topic SurfaceOfRevolution2
Related topic VolumeOfSolidOfRevolution