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Pappus Theorem

Pappus's Theorem
Let  
\[B\]
  be a uniform density region entirely above pr below the  
\[x\]
  axis. If  
\[B\]
  is rotated about the  
\[x\]
  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  
\[B\]
  about the  
\[x\]
  axis is
\[\begin{equation} \begin{aligned} V &= \int^b_a \pi y^2_2 \: dx - \int^b_a \pi y^2_1 \: dx \\ &= - \pi (\int^b_a y^2_1 \: dx + \int^a_b y^2_2 \: dx) \\ &= - \pi ( \int_{C_1} y^2 \: dx + \int_{C_2} y^2 \: dx ) \\ &= - \pi \oint_C y^2 \: dx\end{aligned} \end{equation}\]

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  
\[V= 2 \pi S y_c\]