Area of a Surface of Revolution

A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis . Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis. A circle that is rotated about a diameter generates a sphere, and if the circle is rotated about a co-planar axis other than the diameter it generates a torus.

Surface of Revolution

A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).

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

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

provided that $x(t)$ is never negative between the endpoints $a$ and $b$. The quantity $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2$ comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:

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

If the curve is described by the function $y = f(x)$, $a \leq x \leq b$, then the integral becomes: 

$A_x = 2\pi\int_a^b y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx \\ \quad= 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ 

for revolution around the $x$-axis, and

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

for revolution around the $y$-axis ($a \leq y \leq b$).

Example

The spherical surface with a radius $r$ is generated by the curve $x(t) =r \sin(t)$, $y(t) = r \cos(t)$, when $t$ ranges over $[0,\pi]$. Its area is therefore:

$\begin{aligned} A &{}= 2 \pi \int_0^\pi r\sin(t) \sqrt{\left(r\cos(t)\right)^2 + \left(r\sin(t)\right)^2} \, dt \\ &{}= 2 \pi r^2 \int_0^\pi \sin(t) \, dt \\ &{}= 4\pi r^2 \end{aligned}$