Definitions

# Surface of revolution

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.

Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle that is rotated about a (coplanar) axis through the center generates a sphere. If the axis is coplanar and outside the circle it generates a toroidal surface.

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

$A = 2 pi int_a^b x\left(t\right) sqrt\left\{left\left(\left\{dx over dt\right\}right\right)^2 + left\left(\left\{dy over dt\right\}right\right)^2\right\} , dt,$

provided that $x\left(t\right)$ is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

$left\left(\left\{dx over dt\right\}right\right)^2 + left\left(\left\{dy over dt\right\}right\right)^2$

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity $2pi x\left(t\right)$ is the path of (the centroid of) this small segment, as required by Pappus's theorem.

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

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

for revolution around the x-axis, and

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

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 $\left[0,pi\right]$. Its area is therefore

$A = 2 pi int_0^pi sin\left(t\right) sqrt\left\{left\left(cos\left(t\right)right\right)^2 + left\left(sin\left(t\right)right\right)^2\right\} , dt = 2 pi int_0^pi sin\left(t\right) , dt = 4pi.$

For the case of the spherical curve with radius r, $y\left(x\right) = sqrt\left\{r^2 - x^2\right\}$ rotated about the x-axis

$A = 2 pi int_\left\{-r\right\}^\left\{r\right\} sqrt\left\{r^2 - x^2\right\},sqrt\left\{1 + frac\left\{x^2\right\}\left\{r^2 - x^2\right\}\right\},dx$
$= 2 pi int_\left\{-r\right\}^\left\{r\right\} r,sqrt\left\{r^2 - x^2\right\},sqrt\left\{frac\left\{1\right\}\left\{r^2 - x^2\right\}\right\},dx$
$= 2 pi int_\left\{-r\right\}^\left\{r\right\} r,dx$
$= 4 pi r^2,$

## Rotating a function

To generate a surface of revolution out of any 2-dimensional scalar function $y=f\left(x\right)$, simply make $u$ the function's parameter, set the axis of rotation's function to simply $u$, then use $v$ to rotate the function around the axis by setting the other two functions equal to $f\left(u\right)sin v$ and $f\left(u\right)cos v$ conversely. For example, to rotate a function $y=f\left(x\right)$ around the x-axis starting from the top of the $xz$-plane, parameterize it as $vec r\left(u,v\right)=langle u,f\left(u\right)sin v,f\left(u\right)cos vrangle$ for $uin x$ and $vin\left[0,2pi\right)$ .

## 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.