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

y = x 2

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

0 = x0 < x1 < x2 < ...< xn = 1

and constructing vertical rectangles to approximate the area under the curve. The ith rectangle has height

hi = |\x*i/|2, xi-1 < x*i < xi

and width

wi = /_\ xi = xi- xi-1

The ith rectangle, when revolved about the x-axis, generates a circular disk with volume

2 *4 phiwi = p |\xi /| /_\ xi

as shown in Figure 2.

Figure 2

The approximate volume of the solid is just the sum of the volumes of the circular disks,

n sum *4 p |\xi /| /_\ xi. i=1

The exact volume of the solid, of course, is given by the definite integral

integral 1 4 p- 0 p x dx = 5.

Volume using cylindrical shells

Partition the interval [0,1] on the y-axis into n subintervals by choosing points

0 = y0 < y1 < y2 < ...< yn = 1

and constructing horizontal rectangles to approximate the area under the curve. The ith rectangle has height

h = /_\ y = y - y i i i i-1

and width

V~ -* * wi = 1 - yi , yi- 1 < yi < yi.

The ith rectangle, when revolved about the x-axis, generates a cylindrical shell with volume

V~ --- 2py* wihi = 2py* | \1- y*/| /_\ yi, i i i

as shown in Figure 3.

Figure 3

The approximate volume of the solid is just the sum of the volumes of the cylindrical shells,

sum n * V~ -*- 2pyi| \1- yi /| /_\ yi. i=1

The exact volume of the solid, of course, is given by the definite integral

integral 1 V~ -- p- 2py\|1- y/|dy = 5. 0