The document is notes from a math class on double integrals over general regions. It includes announcements about office hours and problem sessions. It defines double integrals over general regions as limits of integrals over unions of rectangles approximating the region. It discusses properties of double integrals and iterated integrals over curved regions of Type I and Type II. It provides examples and worksheets for students to practice evaluating double integrals.

1. Section 12.3
Double Integrals over General Regions
Math 21a
March 19, 2008
Announcements
◮ Office hours Tuesday, Wednesday 2–4pm SC 323
◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b
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Image: Flickr user Netream
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2. Announcements
◮ Office hours Tuesday, Wednesday 2–4pm SC 323
◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b
. . . . . .

3. Outline
Last Time
Double Integrals over General Regions
Again a Limit
Properties of Double Integrals
Iterated Integrals over Curved Regions
Regions of Type I
Regions of Type II
. . . . . .

4. Definition
The double integral of f over the rectangle R is
∫∫ ∑∑
m n
f(x, y) dA = lim f(x∗ , y∗ ) ∆A
ij ij
m,n→∞
R i=1 j=1
For continuous f this limit is the same regardless of method for
choosing the sample points.
. . . . . .

5. Fubini’s Theorem
Theorem (Fubini’s Theorem)
If f is continuous on R = [a, b] × [c, d], then
∫∫ ∫ b∫ d ∫ d∫ b
f(x, y) dA = f(x, y) dy dx = f(x, y) dx dy
a c c a
R
This is also true if f is bounded on R, f is discontinuous only on a finite
number of smooth curves, and the iterated integrals exist.
. . . . . .

6. Outline
Last Time
Double Integrals over General Regions
Again a Limit
Properties of Double Integrals
Iterated Integrals over Curved Regions
Regions of Type I
Regions of Type II
. . . . . .

7. Towards an integral over general regions
◮ Right now we can integrate over a rectangle
◮ Extend this to an integral over a union of rectangles possibly
overlapping:
∫∫ ∑ ∫∫
n
f(x, y) dA = f(x, y) dA
D1 ∪...∪Dn i=1 D
i
◮ Define the integral over a general region as
∫∫ ∫∫
f(x, y) dA = lim f(x, y) dA
R D1 ∪...∪Dn
where the limit is taken over all unions of rectangles
approximating R
. . . . . .

8. Properties of Double Integrals
∫∫ ∫∫ ∫∫
(a) [f(x, y) + g(x, y)] dA = f(x, y) dA + g(x, y) dA
∫∫
D ∫∫ D D
(b) cf(x, y) dA = c f(x, y) dA
D D
(c) If∫(x, y) ≥ g(x, y) ∫ ∫ all (x, y) ∈ D, then
f∫ for
f(x, y) dA ≥ g(x, y) dA.
D D
(d) If D = D1 ∪ D2 , where D1 and D2 do not overlap except possibly
on ∫
∫ their boundaries,∫
∫ then ∫∫
[f(x, y)] dA = f(x, y) dA + f(x, y) dA
∫∫
D D1 D2
(e) dA is the area of D, written A(D).
D
(f) If m ≤ f(x, y∫ ∫ M for all (x, y) ∈ D, then
)≤
m · A(D) ≤ f(x, y) dA ≤ M · A(D).
. . . . . .

9. Outline
Last Time
Double Integrals over General Regions
Again a Limit
Properties of Double Integrals
Iterated Integrals over Curved Regions
Regions of Type I
Regions of Type II
. . . . . .

10. Definition
A plane region D is said to be of Type I if it lies between the graphs
of two continuous functions of x:
D = { (x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) }
. . . . . .

11. Definition
A plane region D is said to be of Type I if it lies between the graphs
of two continuous functions of x:
D = { (x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) }
Question
What rectangular approximations for such a D would be good in
estimating an integral over D?
. . . . . .

12. Fact
If D is a region of Type I:
D = { (x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) }
Then for any “mostly” continuous function f
∫∫ ∫ b∫ g2 (x)
f(x, y) dA = f(x, y) dy dx
D a g1 (x)
. . . . . .

13. Worksheet #1
Problem ∫
∫ 1 ey √
Evaluate x dx dy
0 y
. . . . . .

14. Worksheet #1
Problem ∫
∫ 1 ey √
Evaluate x dx dy
0 y
Answer
( )
4
−8 + 5e3/2
45
. . . . . .

15. Worksheet #2
Problem ∫
∫
2y √
Evaluate dA, where D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ x}.
x2 +1
D
. . . . . .

16. Worksheet #2
Problem ∫
∫
2y √
Evaluate dA, where D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ x}.
x2 +1
D
Answer
1
ln 2
2
. . . . . .

17. Definition
A plane region D is said to be of Type II if it lies between the graphs
of two continuous functions of y:
D = { (x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y) }
. . . . . .

18. Definition
A plane region D is said to be of Type II if it lies between the graphs
of two continuous functions of y:
D = { (x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y) }
Question
What rectangular approximations for such a D would be good in
estimating an integral over D?
. . . . . .

19. Fact
If D is a region of Type II:
D = { (x, y) | c ≤ x ≤ d, h1 (y) ≤ x ≤ h2 (y) }
Then for any “mostly” continuous function f
∫∫ ∫ d∫ h 2 (y )
f(x, y) dA = f(x, y) dx dy
D c h 1 (y )
. . . . . .

20. Worksheet #3
Problem ∫
∫
Evaluate xy2 dA, where D is bounded byy = x, and x = y2 − 2.
D
. . . . . .

21. Worksheet #3
Problem ∫
∫
Evaluate xy2 dA, where D is bounded byy = x, and x = y2 − 2.
D
Answer
9
7
. . . . . .

22. Worksheet #4
Problem
Find the volume of the solid under the surface z = xy and above the
triangle with vertices (1, 1), (4, 1), and (1, 2).
. . . . . .

23. Worksheet #4
Problem
Find the volume of the solid under the surface z = xy and above the
triangle with vertices (1, 1), (4, 1), and (1, 2).
Answer
∫ 4 ∫ 2−1/3(x−1)
31
xy dy dx =
1 1 8
. . . . . .

24. Worksheet #5
Problem ∫ 4∫
√
x
Sketch the region of integration for f(x, y) dy dx and change the
0 0
order of integration.
. . . . . .

25. Worksheet #5
Problem ∫ 4∫
√
x
Sketch the region of integration for f(x, y) dy dx and change the
0 0
order of integration.
Answer
The integral is equal to
∫ 2 ∫ y2
f(x, y) dx dy
0 0
. . . . . .