16 MULTIPLE INTEGRALS

CHAPTER 16
MULTIPLE INTEGRALS
GROUP WORK 3: Bounding on a Disk
You may or may not tell your students that there is no such thing as Nentebular science.
1
x 2 + y 2 lends itself naturally to this type of analysis, as seen by its graph.
+ 1
The surface
Before handing out this activity, review the method of finding the area of an annulus. Close the activity by
pointing out that if we continued the process of dividing D into smaller and smaller rings, the upper and lower
bounds would approach each other, converging on the actual area:
∫ 2π ∫ 5
0
0
1
r 2 rdrdθ = π ln 2 + π ln 13 ≈ 3.2581π
+ 1
Answers:
1. Answers will vary here. The upper bound should be no worse than 25π ≈ 78.5 (setting the function equal
to the constant 1, its maximum) and the lower bound should be no worse than 25π
26
≈ 3.02 (setting the
function equal to the constant
26 1 , its minimum).
( ) ( )
( )
2. Upper: 4π (1) + (25π − 4π) 15
= 41 5 π ≈ 25.8. Lower: 4π 15
+ (25π − 4π) 126
= 209
130
( )
( )
π ≈ 5.05.
3. Upper: π (1) + (9π − π) 12
+ (25π − 9π) 110
= 33
5 π ≈ 20.7.
( )
( )
( )
Lower: π 12
+ (9π − π) 110
+ (25π − 9π) 126
= 249
130 π ≈ 6.02.
HOMEWORK PROBLEMS
Core Exercises: 1, 5, 10, 13, 22, 32, 43, 45, 51
Sample Assignment: 1, 5, 8, 10, 13, 17, 20, 22, 25, 29, 32, 33, 40, 43, 45, 51, 53, 58, 59
Exercise D A N G
1 ×
5 ×
8 ×
10 ×
13 ×
17 ×
20 ×
22 ×
25 ×
29 ×
Exercise D A N G
32 ×
33 ×
40 ×
43 ×
45 ×
51 × ×
53 ×
58 × ×
59 ×
888