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

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