16 MULTIPLE INTEGRALS

CHAPTER 16
MULTIPLE INTEGRALS
• Present some different uses of polar integrals:
1. Compute ∫∫ (
R x 2 + y 2) 2 dA,whereR is the region enclosed by x 2 + y 2 = 4 between the lines y = x
and y =−x.
y
2
_2
_1
0 1
2
x
_2
2. Compute the volume between the cone z 2 = x 2 + y 2 and the paraboloid z = 4 − x 2 − y 2 .
GROUPWORK1:ThePolarAreaFormula
Perhaps start by having students guess the final answer to Problem 1. Notice that the students will have to be
able to antidifferentiate cos 2 θ and sin 2 θ to complete this activity.
Answers:
1. 2 ∫ π
0 1 2 (1 + cos θ)2 dθ = 3π 2
2. ∫ π
0
[
12
(1 + sin θ) 2 − 1 2 (1)2] dθ = π 4 + 2
GROUP WORK 2: Fun with Polar Area
Many students will write ∫ 3π ∫ θ
0 0
rdrdθ. Try to get them to see for themselves that they must now subtract
the area of the region that is counted twice in this expression.
Answers:
1. 9 2 π3 − 19
6 π3 = 4 3 π3 2. ∫ π
0
∫ 2
0 e−r 2 rdrdθ = π (
2 1 − e
−4 )
GROUP WORK 3: Fun with Polar Volume
If the students get stuck on Problem 2, point out that the intersection can be shown to be x 2 + y 2 = 3 4 , z = 1 2 ,
and hence the integral is over the region x 2 + y 2 ≤ 3 4
. If time is an issue, the students can be instructed to set
up the integrals without solving them.
Answers:
1. ∫ 2π
0
2. ∫ 2π
0
∫ 1
(√
0 1 − r 2 + 1)
rdrdθ = 5π 3
∫ 3/4
[√
0 1 − r 2 −
(1 − √ )]
(
1 − r 2 rdrdθ =
48
π 37 − 7 √ )
7
894