Theory

Theory 1

The infinitesimal of arc length along a curve, $ds$, can be used to find the surface area of a surface of revolution. The circumference of an infinitesimal band is $2\pi R$ and the width of such a band is $ds$.

The general formula for the surface area is:

$$S={\int }_a^{b}2\pi Rds$$

In any given problem we need to find the appropriate expressions for $R$ and $ds$ in terms of the variable of integration. For regions rotated around the $x$-axis, the variable will be $x$; for regions rotated about the $y$-axis it will be $y$.

Assuming the region is rotated around the $x$-axis, and the cross section in the $xy$-plane is the graph of $f$ and so $R=f(x)$, the formula above becomes:

Area of revolution formula - thin bands

The surface area $S$ of the surface of revolution given by $R=f(x)$ is given by the formula:

$$S={\int }_a^{b}2\pi f(x)\sqrt{1+(f^{\prime }{)}^2}dx$$

In this formula, we assume $f(x)\ge 0$ and $f^{\prime }$ is continuous. The surface is the revolution of $f(x)$ on $x\in [a,b]$ around the $x$-axis.