Refresher - Surface Area of a Solid Revolution

A surface of revolution is formed when a curved is rotated about a line. In the case when ${\displaystyle f}$ is positive and has a continuous derivative, we define the surface area of the surface obtained by rotating the curve , about the ${\displaystyle x}$-axis as

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

If the curve described as , then the formula for surface area becomes

${\displaystyle S={\int }_c^{d}2\pi y\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}dy}$

This can be summarized as follows:

${\displaystyle S=\int 2\pi yds,}$

 

where

${\displaystyle ds=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}$ or ${\displaystyle ds=\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}.}$

 

If rotation is about the y-axis then surface area is:

${\displaystyle S=\int 2\pi xds,}$

 

where

${\displaystyle ds=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}$ or ${\displaystyle ds=\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}}$