DIFFERENTIAL AND INTEGRAL CALCULUS. 103 



Also PK= ~ = T-^-I = MP + 2rt > 



smip 1 smy 



i.e. the normal intercepted by the axis of y exceeds the 

 abscissa by a constant quantity. The radius of curvature 



2a PK* 



For the area of any part of the loop, 

 2 r* /(x + a\ 



= ^L u v\~~^J^ a ~ x ' )0ri 



Q /q 



hence the area of the whole loop is 2 . 



5 



The volume generated by the revolution of the loop 

 = . T (^ + a)(x- 



Ja J -a 



6.27 



Also the area of the surface generated by the revolution 

 of the loop 



ds 



7 



= 2-7T v -r-dx, 

 J dx 



ds 1 



therefore y = - . 



y d x 6 



therefore 



area 



2-7T ~ (4 - ") ^ = TT ^ {12 - 3} = 



