# Revolving a region about the x-axis

The region in the first quadrant bounded by the x-axis, the line x = 1 and the parabola

is revolved about the x-axis, generating the solid shown in Figure 1.

 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 x-axis.

### Volume using circular disks

Partition the interval [0,1] on the x-axis 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 x-axis, generates a circular disk with volume

as shown in Figure 2.

 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

### Volume using cylindrical shells

Partition the interval [0,1] on the y-axis 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 x-axis, generates a cylindrical shell with volume

as shown in Figure 3.

 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