Answer to each blank to correctly complete the explanation for deriving the formula for the volume of a sphere.
For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius r and height ________ minus the volume of two cones, each with a radius and height of r. A cross section of the sphere is __________, and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is __________.
The volume of the cylinder with radius r and height 2r is 2πr3
, and the volume of each cone with radius r and height r is _________ . So the volume of the cylinder minus the two cones is 43πr3 . Therefore, the volume of the cylinder is ___________ by Cavalieri's principle.
( possible Answers to the blanks)
r/2 , r , 2r , an annulus a circle , 1/3(pi)r^3 , 2/3(pi)r^3 , 4/3(pi)r^3 , 5/3(pi)r^3 ,
2(pi)r^3 , 4(pi)r^3
