9 FURTHER APPLICATIONS OF INTEGRATION

SECTION 9.2
AREA OF A SURFACE OF REVOLUTION
the points (0, 1) and (1, 0), and then the cone generated by the line segment of slope m =−1 through the
( ) 1
point
2 3/2 , 1
2 3/2 .
• Compute the surface area when the function f (x) = 2 3 x3/2 ,1≤ x ≤ 2 is rotated about the y-axis.
• Show that the area of the surface of revolution formed when f (x) = x 4 is rotated about the x-axis cannot
be directly computed. Show that this is also true when the function is rotated about the y-axis.
GROUP WORK 1: Gabriel’s Horn
There are two forms of this exercise. Some groups should be given the first, some the second. The students
should not be told that there are two different forms. After they have made their conclusions, poll the class.
Then have the students who think that the horn can be painted try to defend their case, and the ones who say
it cannot defend theirs. Close by having the students try to understand how a surface with infinite area can
contain a finite volume (see Exercise 25).
Answers:
∫ ∞
( ) 1 2
Form One π dx = π ft 3
1 x
∫ ∞
√
2π
Form Two
1 + 1 ∫ ∞
x x 4 dx diverges, by comparison to 2π
x dx
1
GROUP WORK 2: Mind Your p’s and q’s
This activity involves some subtle comparisons and proves an important mathematical point.
Answers:
1. 2π
√
( p
) 2
∫
4π ∞
x p 1 + <
x p+1 x p and 4π
dx converges.
1 x p
2. 2π
√
( q ) 2
∫
2π ∞
x q 1 + >
x q+1 x q and 2π
dx diverges.
xq HOMEWORK PROBLEMS
Core Exercises: 1, 6, 15, 17, 28
Sample Assignment: 1, 6, 9, 12, 14, 15, 17, 19, 22, 25, 28, 33
1
1
Exercise D A N G
1 ×
6 ×
9 ×
12 ×
14 ×
15 ×
Exercise D A N G
17 × ×
19 × ×
22 ×
25 ×
28 × ×
33 ×
497