Revolving a region about the y-axis

The region in the first quadrant bounded by the axes and the parabola

y = 1- x2

is revolved about the y-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 y-axis.

Volume using circular disks

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

hi = /_\ yi = yi- yi-1

and width

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

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

p w2ihi = p|\1- y*i /| /_\ yi

as shown in Figure 2.

Figure 2

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

sum n * p|\1 - yi /| /_\ yi. i= 1

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

integral 1 p- p\|1 - y/|dy = 2. 0

Volume using cylindrical shells

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

h = 1 - x*2, x < x* < x . i i i-1 i i

and width

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

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

2px*i hiwi = 2px*i| \1 - x*i 2/| /_\ xi,

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 * *2 2px i|\1 - xi /| /_\ xi. i= 1

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

integral 1 2 p- 2px|\1 - x /|dx = 2 . 0