## Pappus Theorem

Pappus's Theorem
Let
$B$
be a uniform density region entirely above pr below the
$x$
axis. If
$B$
$x$
axis then the volume of the solid generated is
$V= 2 \pi S y_c$
where
$S$
is the area of
$B$
and
$(x_c , y_c )$
is the centroid of
$B$

Proof The volume of the solid generated by revolving
$B$
$x$
axis is
\begin{aligned} V &= \int^b_a \pi y^2_2 \: dx - \int^b_a \pi y^2_1 \: dx \\ &= - \pi (\int^b_a y^2_1 \: dx + \int^a_b y^2_2 \: dx) \\ &= - \pi ( \int_{C_1} y^2 \: dx + \int_{C_2} y^2 \: dx ) \\ &= - \pi \oint_C y^2 \: dx\end{aligned}

Apply Green's Theorem with
$P= \pi y^2, \: Q=0$
to give
$V = - \pi \oint_C y^2 \: dx =2 \pi \int_B y \: dx \; dy$

The centre of mass of the centroid is
$(x_c , y_c ) =(\frac{ \int_B x \rho \: dx \: dy}{M} , \frac{ \int_B y \rho \: dx \: dy}{M} ) =(\frac{\int_B x \: dx \:dy}{S} , \frac{\int_B y \: dx \:dy}{S} )$

if the lamina has uniform density.
Then
$V= 2 \pi S y_c$