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 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

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