the region between two concentric circles is called an annulus, or more informally, a washer
DESCRIPTION
A solid of revolution is a solid obtained by rotating a region in the plane about an axis. The sphere and right circular cone are familiar examples of such solids. Each of these is “swept out” as a plane region revolves around an axis. - PowerPoint PPT PresentationTRANSCRIPT
A solid of revolution is a solid obtained by rotating a region in the plane about an axis. The sphere and right circular cone are familiar examples of such solids. Each of these is “swept out” as a plane region revolves around an axis.
Volume of Revolution: Disk Method If f (x) is continuous and f (x) ≥ 0 on [a, b], then the solid obtained by rotating the region under the graph about the x-axis has volume [with R = f (x)]
22b b
a a
V R dx f x dx
Calculate the volume V of the solid obtained by rotating the region under y = x2 about the x-axis for 0 ≤ x ≤ 2.
22b b
a a
V R dx f x dx
22 522
0 0
5
5
2 325 5
xx dx
The region between two concentric circles is called an annulus, or more informally, a washer.
2 22 2outer inner
b b
a a
V R R dx f x g x dx
Region Between Two Curves Find the volume V obtained by revolving the region between y = x2 + 4 and y = 2 about the x-axis for 1 ≤ x ≤ 3.
2 2b
a
V f x g x dx
3 3
22 2 4 2
1 1
35 3 5
1
4 2 8 12
8 3 1 812 72 36 125 3 5
2126155 3
V x dx x x dx
x x x
Revolving About a Horizontal Axis Find the volume V of the “wedding band” obtained by rotating the region between the graphs of f (x) = x2 + 2 and g (x) = 4 − x2 about the horizontal line y = − 3.
12 22 2
1
12 22 2
0
4 3 2 3
2 7 5 32
V x x dx
x x dx
2 2b
a
V f x g x dx
Find the volume obtained by rotating the graphs of f (x) = 9 − x2 and y = 12 for 0 ≤ x ≤ 3 about (a) the line y = 12 (b) the line y = 15.
2b
a
V f x dx
3 22
0
12 9 6485
V x dx
3 22 2
0
322
0
11
15 9 3
86 895
V x dx
x dx
Find the volume obtained by rotating the graphs of f (x) = 9 − x2 and y = 12 for 0 ≤ x ≤ 3 about (a) the line y = 12 (b) the line y = 15.
2 2b
a
V f x g x dx
We can use the disk and washer methods for solids of revolution about vertical axes, but it is necessary to describe the graph as a function of y—that is, x = g (y).Revolving About a Vertical Axis Find the volume of the solid obtained by rotating the region under the graph of f (x) = 9 − x2 for 0 ≤ x ≤ 3 about the vertical axis x = −2.
2
outer inner
9 for 0 3 9 (since 0)
9 2, 2
y x x x y x
R y R
2 2d
c
V f y g y dy