APPLICATIONS OF INTEGRATION
6.2 Volumes
Objective: Find volume of solids of revolution
I. Definition of volume
A. S is a solid which lies between x = a and x = b
B. Cross-sectional area of S in plane P x, through x and perpendicular to x-axis is A(x)
C. A is a continuous function
D. nb
*
ia
ni1
VlimA(x)xA(x)dx
→∞ =
=∆=
∑∫
II. A solid of revolution is formed by revolving a region in a plane about a line in the plane,
called the axis of revolution.
III. Disk method: Find the volume of the solid of
revolution generated by revolving the region formed
by x = 0, x = 3, and the
32
1
yx2xx4,
2
=−++
x-axis about the x–axis.
A. Consider a regular partition on [0, 3] with 1
x4
∆=
B. Rotate area about x–axis
C. Each rectangle generates a circular disk
D. Volume of a disk = 2
rh
π
E. Radius of a disk = f (x)
F. Height (thickness) of a disk = x∆
G. Volume of a disk = (thickness)
2
(radius)π
H. Volume of a disk = 2
32
1
x2xx4x
2
π
−++∆
I.
222222
2
L
123456
Vf(0)ffffff
444444
π
=++++++
+
22222
78910111
fffff
444444
++++
LRM
V28.35,V25.91,V27.10
πππ
≈≈≈
