DIFFERENTIAL AND INTEGRAL CALCULUS. 105 



be perpendicular to the axis of y, 



PK=NP+2a: and if KPN= 0. NP= * a co * 6 



l-cos0 



4. If s be the arc of a curve measured from any fixed 

 point to the point (a?, y) or (r, 0), then will 



Prove that the whole arc of the curve x% + y* = a^ is 6<z, 

 and that of the curve in (3) measured from the vertex 

 ( VBP) is equal to OK. 



5. Find the differential coefficients of the area of any 

 curve (1) included between the curve, the axis of x, and 

 two ordinates; (2) between the curve and two radii 



vectores. The area of the loop of the curve y* = x* - 



a -\-x 



is a 2 (2 JTT) ; and the whole area of the curve r 2 = a* cos 20 

 is a 2 . 



6. Find the differential coefficients of the volume, and 

 of the area of the surface generated by the revolution of 

 a given curve about the axis of x. If the curve be that 

 in (3), the volume generated by the loop is f Tra 3 , and the 

 area of the surface generated is 3 Tra 2 . 



7. The origin of polar coordinates is S, SA is (fig. 28) 

 a fixed radius vector, P any point of a curve, SP=r, 

 ASP=0) and U the volume generated by the revolution 



of ASP about 8A ; prove that -^ = f^ 3 sin0. 



The volume of the spherical sector included between 

 a sphere and a cone of vertical angle 2a, with its vertex on 

 the circumference, and its axis passing through the centre 



8. If NP (fig. 29) be the ordinate, PG the normal, 

 terminated by the axis of x at any point P of a curve, and 



p 



