16 MULTIPLE INTEGRALS

SECTION 16.3
DOUBLE INTEGRALS OVER GENERAL REGIONS
WORKSHOP/DISCUSSION
• Let D be the region shown below. Set up ∫∫ D f (x, y) dA both as a type I integral [∫∫ D
and a type II integral [∫∫ D f (x, y) dydx] .
f (x, y) dx dy]
y
6
4
2
D
y=2x
_1 1 2 3
x
_2
x+1=(y-1)$
• Evaluate ∫∫ D yx2 dA,whereD is the unit circle, both as a type I integral and as a type II integral.
• Show how to change order of integration in ∫ 1
• Change order of integration for ∫ 1
0
GROUP WORK 1: Type I or Type II?
0
∫ y 1/3
(2/π) arcsin y
∫ x 1/3
x 4
f (x, y) dydx.
f (x, y) dx dy.
Before handing out this activity, remind the students of the definitions of type I and type II regions given in
the text.
Answers:
1. Both 2. Both 3. Type II 4. Neither 5. Type II 6. Type I 7. Both 8. Type II 9. Neither 10. Both
GROUP WORK 2: Fun with Double Integrals
In this section, it is probably best to have the students get experience in setting up many double integrals,
rather than have them spend a lot of time solving only one. During the homework, of course, both skills
should be practiced.
Answers:
1. (a) ∫ 3/4
0
∫ x−x 2
x/4
dydx. This can also be done as two integrals in dx dy.
(b) ∫ 1 ∫ √ 4 x
0
dydx or ∫ 1 ∫ √ y
x 2 0
dx dy
y 4
(c) ∫ 1/2 ∫ √ 4 y
0
dx dy. This can also be done as two integrals in dydx.
y 2
(d) ∫ 1 ∫ √ 2− y
√
0 y
dx dy. This can also be done as two integrals in dydx.
2. A hemisphere
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