144 CENTER OF GRAVITY 



19. Find the center of gravity of a lamina bounded by a parabola and 

 a line perpendicular to its axis. 



20. Find the center of gravity of the volume cut from a solid parabo- 

 loid by a plane perpendicular to its axis. 



21. Find the center of gravity. of the area inclosed by two radii of 

 an ellipse. 



22. Find the center of gravity of the volume cut off from a solid ellipsoid 

 by a plane through the center. 



23. Find the center of gravity of half of an ellipsoidal shell, this being 

 bounded by two similar concentric and coaxial ellipsoids, and a plane 

 through the center. 



24. A right circular cone whose base is of radius r is divided into two 

 equal parts by a plane through the axis. Prove that the distance of the 



center of gravity of either half from the axis is 



it 



25. Find the center of gravity of a lamina bounded by the semicubical 

 parabola a; 3 = ay 2 , the axis of x, and the ordinate x = a. 



26. Find the center of gravity of a single loop of the curve 



r = a sin 30. 



27. Find the center of gravity of an octant of a sphere. 



28. A cylindrical hole of radius a is drilled through a hemisphere of 

 radius b so that the radius perpendicular to the base of the hemisphere is 

 also the central line of the hole. Find the center of gravity of the figure. 



29. Find the center of gravity of the area inclosed between the two 

 circles a? + yi = fl2 . x i + y i = 2 ab. 



30. Find the center of gravity of a lens made of homogeneous glass, 

 having spherical surfaces of radii r, s, and of which the thickness is t at 

 the center and zero at the edge. 



