16 MULTIPLE INTEGRALS

( ∫ 1 ∫ √ 4−x 2
0 1
SECTION 16.6
4. Symmetry gives 4
( ∫ 1 ∫ √ 4−x
8
2 ∫ √ 4−x 2 −z 2
0 1 0
dydzdx + ∫ 2 ∫ √ 4−x 2
1 0
TRIPLE INTEGRALS
∫ √
√
4−x2−z2
dydzdx + ∫ 2
− 4−x 2 −z 2 1
∫ √ 4−x 2
0
)
∫ √ 4−x 2 −z 2
0
dydzdx
∫ √ )
4−x 2 −z
√
2
dydzdx
− 4−x 2 −z 2
or
GROUP WORK 3: An Unusual Volume
This is a challenging group work for more advanced students. The idea is to show that just because a solid
looks simple, the computation of its volume may be difficult. The line generated by P 1 and P 2 has equation
z =− 3 4
x + 4, and hence this equation, interpreted in three dimensions, is also the equation of the plane S.
The integral V (E) = ∫ 2 ∫ √ ( )
√
4−(x−2) 2
−2
− 3
− 4−(x−2) 2 4 x + 4 dydx requires polar coordinates to solve by hand (since
the bounding circle has equation r = 4cosθ, 0≤ θ ≤ π) and also requires the students to remember how to
integrate cos 2 θ and cos 4 θ. The volume is 16π.
Note that the problem can be simplified by moving the solid so that the z-axis runs through the center of D.
Point out that a simple geometric solution can be obtained by replacing S by the horizontal plane z = 5 2 , thus
giving a standard cylinder.
HOMEWORK PROBLEMS
Core Exercises: 5, 11, 13, 19, 24, 25, 27, 34, 39
Sample Assignment: 2, 5, 8, 10, 11, 13, 17, 19, 21, 23, 24, 25, 26, 27, 30, 34, 39, 42, 46, 50, 51
Exercise D A N G
2 ×
5 ×
8 ×
10 ×
11 ×
13 ×
17 ×
19 ×
21 ×
23 ×
24 ×
Exercise D A N G
25 ×
26 ×
27 ×
30 ×
34 × ×
39 ×
42 ×
46 × ×
50 ×
51 ×
907