In Example 5.5.1, the surface area of the given surface of revolution was found to be

In Example 5.7.7, the $y$-coordinate of the centroid of the curve defined by $y\=x^2\,x\in \left[0\,1\right]$ was found to be

$\stackrel{\&conjugate0;}{y}\= \frac{-\frac{1}{32}\ln \left(2\right)\+\frac{9}{32}\sqrt{5}-\frac{1}{64}\ln \left(\frac{1}{2}\+\frac{1}{4}\sqrt{5}\right)}{\frac{1}{2}\sqrt{5}-\frac{1}{4}\ln \left(-2\+\sqrt{5}\right)}\doteq 0.4099802175$ 

From Table 5.4.1, the length of the given curve is given by

$L\={\int }_0^1\sqrt{1\+4 x^2}\mathit{\ⅆ}x\=\frac{\sqrt{5}}{2}-\frac{\ln \left(\sqrt{5}-2\right)}{4}\doteq 1.478942857$ 

Hence, the second theorem of Pappus gives, for the surface area of the given surface of revolution,

$2 \mathrm{\π} \stackrel{\&conjugate0;}{y} L\=2 \mathrm{\π} \left(-\frac{1}{32}\ln \left(2\right)\+\frac{9}{32}\sqrt{5}-\frac{1}{64}\ln \left(\frac{1}{2}\+\frac{1}{4}\sqrt{5}\right)\right)\doteq 3.809729704$$$