area of surface of revolution

A surface of revolution 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 coordinates, and specializes to the case of a surface of revolution generated by rotating about the $x$-axis a curve described by $y$ in the interval $[a,b]$, its area can be calculated by the formula

$$A=2\pi {\int }_a^{b}y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}𝑑x$$

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

$$A=2\pi {\int }_a^{b}x\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}𝑑y$$

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

$$A=2\pi {\int }_a^{b}y\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}𝑑t$$

To obtain a specific surface of revolution, translation 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

$$A=2\pi {\int }_a^b(y-s)\sqrt{{\left(\frac{dy}{dx}\right)}^2}𝑑y$$

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