Curve

Axis of Rotation

$y\=c$

$x\=c$

$y\=f\left(x\right)$

$\mathrm{\ρ}\=\left|y-c\right|$ = $\left|f\left(x\right)-c\right|$

$\mathrm{ds}\=\sqrt{1\+{\left(\frac{\mathrm{df}}{\mathrm{dx}}\right)}^2}\mathrm{dx}$

$\mathrm{\ρ}\=\left|x-c\right|$ 

$\mathrm{ds}\=\sqrt{1\+{\left(\frac{\mathrm{df}}{\mathrm{dx}}\right)}^2}\mathrm{dx}$

$x\=g\left(y\right)$

$\mathrm{\ρ}\=\left|y-c\right|$

$\mathrm{ds}\=\sqrt{1\+{\left(\frac{\mathrm{dg}}{\mathrm{dy}}\right)}^2}\mathrm{dy}$

$\mathrm{\ρ}\=\left|x-c\right|$ = $\left|g\left(y\right)-c\right|$ 

$\mathrm{ds}\=\sqrt{1\+{\left(\frac{\mathrm{dg}}{\mathrm{dy}}\right)}^2}\mathrm{dy}$

$x\=x\left(t\right)$

$y\=y\left(t\right)$

$\mathrm{\ρ}\=\left|y-c\right|$ = $\left|y\left(t\right)-c\right|$ 

$\mathrm{ds}\=\sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^2\+{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^2}\mathrm{dt}$

$\mathrm{\ρ}\=\left|x-c\right|$ = $\left|x\left(t\right)-c\right|$ 

$\mathrm{ds}\=\sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^2\+{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^2}\mathrm{dt}$

Table 5.5.1   Surface area of a surface of revolution: $2 \mathrm{\π}{\int }_a^b\mathrm{\ρ} \mathit{\ⅆ}s$

Example 5.5.1

Calculate the surface area of the surface of revolution formed when the graph of $y\=x^2\,x\in \left[0\,1\right]$, is rotated about the $x$-axis.

Example 5.5.2

Calculate the surface area of the surface of revolution formed when the graph of $y\=x^2\,x\in \left[0\,1\right]$, is rotated about the line $y\=2$.

Example 5.5.3

Calculate the surface area of the surface of revolution formed when the graph of $y\=x^2\,x\in \left[0\,1\right]$, is rotated about the $y$-axis.

Example 5.5.4

Calculate the surface area of the surface of revolution formed when the graph of $y\=x^2\,x\in \left[0\,1\right]$, is rotated about the line $x\=2$.

Example 5.5.5

Calculate the surface area of the surface of revolution formed when the graph of $y\=e^{-x}\,x\in \left[0\,2\right]$, is rotated about the $y$-axis.