-11 ) 



INTEGRAL CALCULUS. 



1013. The area common to two ellipses which have the same 

 and equal axes inclined at an angle a is 



(a o) sin a 

 1014. The arc of the curve 



between the origin and the point where the curve again meets the 

 of Xy is equal to the perimeter of an ellipse of axes 2, 26. 

 Determine the ratio of a : 6 in order that the area included 

 between tin's part of the curve and the axis of x may be equal to 

 the area of the ellipse. 



1015. A series of spheres touch eah otlier at a given point 

 and from each is cut off a segment of gi rface by a 



plan* bbe line oi : prove that the circular 



sections made by these planes lie on the same sphere. 



1016. The sum of tin- pp-dm-is of m.-li clement of an elliptic 



DOe from tli<> focus is equal to 



.If.' " , M being the mass of the lamina, 2a the major axis, 



9 



and /; the rcrc-ntririty. 



1017. If the areas of the curves 



ay (*-&) = (a f -a ) (&* -a 1 )', a^ + y^a', (6>a) 

 be A t A'; prove that, as 6 decreases to a, the limiting value of 

 A'-A 



is 



a 



142 



