INTEGRATION FORMULA 137 



If the solid is homogeneous, p is constant, and the formulae 

 become 



z 

 etc. 



I I \xdxdydz I I \ydxdyd 



% = JJJ _ , y = JJJ 



i I \dxdydz I I \dxdydz 



107. Use of polar coordinates. Any other system of coordinates 

 can, of course, be used for finding a center of gravity by integra- 

 tion. The only coordinates besides Cartesians which are of much 

 use for this purpose are polar coordinates. 



We can find the center of gravity of a lamina in polar coordinates 

 by supposing the Cartesian coordinates x y y connected with the 

 polar coordinates r, 6 by the usual transformation 



x = r cos 6, y = r sin 6. 

 Formulae (31) then become 



rcosfl 



r sin0 = 



CCp (r cos 6) (r drd0) CCpr 2 cos drdd 

 CCp(rdrdO) CCprdrdO 



CCp (r sin 6} (r drdO) CCpr* sin drd0 

 CCp (r drdd) CCpr drdd 



in which r, are the polar coordinates of the center of gravity. 

 On dividing corresponding sides of these equations, we can obtain 

 an equation giving the coordinate alone, namely 



CCpr*s 

 JJ 



sm0drd0 



cos drdd 



