derivation of Pappus’s centroid theorem

I.  Let $s$ denote the arc rotating about the $x$-axis (and its length) and $R$ be the $y$-coordinate of the centroid of the arc.  If the arc may be given by the equation

$$y=y(x)$$

where  $a\le x\le b$, the area of the formed surface of revolution is

$$A=\mathrm{\hspace{0.33em}2}\pi {\int }_a^{b}y(x)\sqrt{1+{[y^{\prime }(x)]}^2}𝑑x.$$

This can be concisely written

$A=\mathrm{\hspace{0.33em}2}\pi {\displaystyle {\int }_s}y𝑑s$(1)

since differential-geometrically, the product $\sqrt{1+{[y^{\prime }(x)]}^2}dx$ is the arc-element.  We rewrite (1) as

$$A=s\cdot 2\pi \cdot \frac{1}{s}{\int }_{s}y𝑑s.$$

Here, the last factor is the ordinate of the centroid of the rotating arc, whence we have the result

$$A=s\cdot 2\pi R$$

which states the first Pappus’s centroid theorem.

II.  For deriving the second Pappus’s centroid theorem, we suppose that the region defined by

$$a\le x\le b,0\le y_1(x)\le y\le y_2(x),$$

having the area $A$ and the centroid with the ordinate $R$, rotates about the $x$-axis and forms the solid of revolution with the volume $V$.  The centroid of the area-element between the arcs  $y=y_1(x)$  and  $y=y_2(x)$  is $[y_2(x)+y_1(x)]/2$ when the abscissa is $x$; the area of this element with the width $dx$ is $[y_2(x)-y_1(x)]dx$.  Thus we get the equation

$$R=\frac{1}{A}{\int }_a^b\frac{y_2(x)+y_1(x)}{2}[y_2(x)-y_1(x)]𝑑x$$

which may be written shortly

$R={\displaystyle \frac{1}{2A}}{\displaystyle {\int }_a^b}(y_2^2-y_1^2)𝑑x.$(2)

The volume of the solid of revolution is

$$V=\pi {\int }_a^b(y_2^2-y_1^2)𝑑x=A\cdot 2\pi \cdot \frac{1}{2A}{\int }_a^b(y_2^2-y_1^2)𝑑x.$$

By (2), this attains the form

$$V=A\cdot 2\pi R.$$