HI 



x. Hence by proceeding as before we shall hare 





rhere y - ^ (*) is the equation to the lower bounding cnnre 

 '*$ f jf) ' I'lKT, and r and A arc the abscissa) of 



.anca which bound the solid of revolution perpend 

 its axia. 



solid is formed by revolving the area in- 

 * ded between two curves round the axis of y t we shall 



rr%** 

 - 



Or we may use polar formulas. Suppose the figur 



revolve round the axia of x ; let r, 6 be the polar co- 



itca of i ; and r -f Ar, 6 -f A^ those of t. The volume 



of the ring generated by the revolution of the area st is ulti- 



Trsin^rArAd; and the abscissa of the centre of 



gravity of the ring is ultimately r cos 0. Hence 





Similarly, if the figure revolve round the axis of y 



_ 



7 



have hitherto assumed the solid of revolution to be 



of uniform di i this be not the case the formulas must 



be modified. For example, take the first formula in the 



e; suppose that p denotes the density at the 



(x, y). Then the mass of the ring considered will be 



rpy Ay Ax. Hence 



