Calculus 2nd Edition Rogawski

SECTION 6.3 Volumes of Revolution 319
Axis
x = −2
y
y
y
9
y = 9 − x 2
y
A
R outer = 9 − y + 2
B
R outer
R inner
A
B
R inner = 2
x
x
−2 0 x 3
−2 0
−2
3
x
FIGURE 9
The region extends from y = 0 to y = 9 along the y-axis, so
V = π
∫ 9
0
(
R
2
outer − R 2 inner)
dy = π
∫ 9
0
(
9 − y + 4 √ )
9 − y dy
= π
(9y − 1 2 y2 − 8 ) ∣ ∣∣∣
9
3 (9 − y)3/2
0
= 225
2 π
6.3 SUMMARY
• Disk method When you rotate the region between two graphs about an axis, the segments
perpendicular to the axis generate disks or washers. The volume V of the solid of
revolution is the integral of the areas of these disks or washers.
• Sketch the graphs to visualize the disks or washers.
• Figure 10(A): Region between y = f (x) and the x-axis, rotated about the x-axis.
– Vertical cross section: a circle of radius R = f (x) and area πR 2 = πf (x) 2 :
V = π
∫ b
a
R 2 dx = π
∫ b
a
f (x) 2 dx
• Figure 10(B): Region between y = f (x) and y = g(x), rotated about the x-axis.
– Vertical cross section: a washer of outer radius R outer = f (x) and inner radius
R inner = g(x):
V = π
∫ b
a
∫
( b
R
2
outer − Rinner) 2 (
dx = π f (x) 2 − g(x) 2) dx
• To rotate about a horizontal line y = c, modify the radii appropriately:
– Figure 10(C): c ≥ f (x) ≥ g(x):
a
R outer = c − g(x),
R inner = c − f (x)
– Figure 10(D): f (x) ≥ g(x) ≥ c:
R outer = f (x) − c,
R inner = g(x) − c
• To rotate about a vertical line x = c, express R outer and R inner as functions of y and
integrate along the y axis.