The region in the first quadrant bounded by the axes and the parabola
is revolved about the yaxis, generating the solid shown in Figure 1.




The volume of this solid can be approximated by first approximating the area of the planar region with rectangles and revolving these rectangles about the yaxis.
Partition the interval [0,1] on the yaxis into n subintervals by choosing points
and constructing horizontal rectangles to approximate the area under the curve. The ith rectangle has height
and width
The ith rectangle, when revolved about the yaxis, generates a circular disk with volume
as shown in Figure 2.




The approximate volume of the solid is just the sum of the volumes of the circular disks,
The exact volume of the solid, of course, is given by the definite integral
Partition the interval [0,1] on the xaxis into n subintervals by choosing points
and constructing vertical rectangles to approximate the area under the curve. The ith rectangle has height
and width
The ith rectangle, when revolved about the yaxis, generates a cylindrical shell with volume
as shown in Figure 3.




The approximate volume of the solid is just the sum of the volumes of the cylindrical shells,
The exact volume of the solid, of course, is given by the definite integral