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Section 12.3Double 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
..Image: Flickr user Netream
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Announcements
◮ Office hours Tuesday, Wednesday 2–4pm SC 323◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b
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Outline
Last Time
Double Integrals over General RegionsAgain a LimitProperties of Double Integrals
Iterated Integrals over Curved RegionsRegions of Type IRegions of Type II
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DefinitionThe double integral of f over the rectangle R is∫∫
R
f(x, y) dA = limm,n→∞
m∑i=1
n∑j=1
f(x∗ij , y∗ij )∆A
For continuous f this limit is the same regardless of method forchoosing the sample points.
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Fubini’s Theorem
Theorem (Fubini’s Theorem)If f is continuous on R = [a, b] × [c, d], then∫∫
R
f(x, y) dA =
∫ b
a
∫ d
cf(x, y) dy dx =
∫ d
c
∫ b
af(x, y) dx dy
This is also true if f is bounded on R, f is discontinuous only on a finitenumber of smooth curves, and the iterated integrals exist.
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Outline
Last Time
Double Integrals over General RegionsAgain a LimitProperties of Double Integrals
Iterated Integrals over Curved RegionsRegions of Type IRegions of Type II
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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: ∫∫D1∪...∪Dn
f(x, y) dA =n∑
i=1
∫∫Di
f(x, y) dA
◮ Define the integral over a general region as∫∫R
f(x, y) dA = lim∫∫
D1∪...∪Dn
f(x, y) dA
where the limit is taken over all unions of rectanglesapproximating R
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Properties of Double Integrals(a)
∫∫D
[f(x, y) + g(x, y)] dA =
∫∫D
f(x, y) dA +
∫∫D
g(x, y) dA
(b)∫∫D
cf(x, y) dA = c∫∫D
f(x, y) dA
(c) If f(x, y) ≥ g(x, y) for all (x, y) ∈ D, then∫∫D
f(x, y) dA ≥∫∫D
g(x, y) dA.
(d) If D = D1 ∪ D2, where D1 and D2 do not overlap except possiblyon their boundaries, then∫∫
D
[f(x, y)] dA =
∫∫D1
f(x, y) dA +
∫∫D2
f(x, y) dA
(e)∫∫D
dA is the area of D, written A(D).
(f) If m ≤ f(x, y) ≤ M for all (x, y) ∈ D, then
m · A(D) ≤∫∫D
f(x, y) dA ≤ M · A(D).
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Outline
Last Time
Double Integrals over General RegionsAgain a LimitProperties of Double Integrals
Iterated Integrals over Curved RegionsRegions of Type IRegions of Type II
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DefinitionA plane region D is said to be of Type I if it lies between the graphsof two continuous functions of x:
D = { (x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x) }
QuestionWhat rectangular approximations for such a D would be good inestimating an integral over D?
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DefinitionA plane region D is said to be of Type I if it lies between the graphsof two continuous functions of x:
D = { (x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x) }
QuestionWhat rectangular approximations for such a D would be good inestimating an integral over D?
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FactIf 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∫∫D
f(x, y) dA =
∫ b
a
∫ g2(x)
g1(x)f(x, y) dy dx
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Worksheet #1
Problem
Evaluate∫ 1
0
∫ ey
y
√x dx dy
Answer445
(−8 + 5e3/2
)
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Worksheet #1
Problem
Evaluate∫ 1
0
∫ ey
y
√x dx dy
Answer445
(−8 + 5e3/2
)
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Worksheet #2
ProblemEvaluate
∫∫D
2yx2 + 1
dA, where D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤√
x}.
Answer12
ln 2
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Worksheet #2
ProblemEvaluate
∫∫D
2yx2 + 1
dA, where D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤√
x}.
Answer12
ln 2
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DefinitionA plane region D is said to be of Type II if it lies between the graphsof two continuous functions of y:
D = { (x, y) | c ≤ y ≤ d, h1(y) ≤ x ≤ h2(y) }
QuestionWhat rectangular approximations for such a D would be good inestimating an integral over D?
. . . . . .
DefinitionA plane region D is said to be of Type II if it lies between the graphsof two continuous functions of y:
D = { (x, y) | c ≤ y ≤ d, h1(y) ≤ x ≤ h2(y) }
QuestionWhat rectangular approximations for such a D would be good inestimating an integral over D?
. . . . . .
FactIf 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
f(x, y) dA =
∫ d
c
∫ h2(y)
h1(y)f(x, y) dx dy
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Worksheet #3
ProblemEvaluate
∫∫D
xy2 dA, where D is bounded byy = x, and x = y2 − 2.
Answer97
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Worksheet #3
ProblemEvaluate
∫∫D
xy2 dA, where D is bounded byy = x, and x = y2 − 2.
Answer97
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Worksheet #4
ProblemFind the volume of the solid under the surface z = xy and above thetriangle with vertices (1, 1), (4, 1), and (1, 2).
Answer ∫ 4
1
∫ 2−1/3(x−1)
1xy dy dx =
318
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Worksheet #4
ProblemFind the volume of the solid under the surface z = xy and above thetriangle with vertices (1, 1), (4, 1), and (1, 2).
Answer ∫ 4
1
∫ 2−1/3(x−1)
1xy dy dx =
318
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Worksheet #5
Problem
Sketch the region of integration for∫ 4
0
∫ √x
0f(x, y) dy dx and change the
order of integration.
AnswerThe integral is equal to ∫ 2
0
∫ y2
0f(x, y) dx dy
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Worksheet #5
Problem
Sketch the region of integration for∫ 4
0
∫ √x
0f(x, y) dy dx and change the
order of integration.
AnswerThe integral is equal to ∫ 2
0
∫ y2
0f(x, y) dx dy
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