Math 251-copyright Joe Kahlig, 15C
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Section 13.3: Double Integrals over General Regions
Example: Evaluate the integral where D = {(x, y) ∈ <2 |x2 + y 2 ≤ 16}.
ZZ p
16 − x2 − y 2 dA
D
Definition: A plane region D is said to be of Type I if it lies between two continuous functions of
x, that is
D = {(x, y)|a ≤ x ≤ b, B(x) ≤ y ≤ T (x)}
Let D = {(x, y)|a ≤ x ≤ b, B(x) ≤ y ≤ T (x)} be a domain of type I and f is continuous on D.
Choose a rectangle region R = [a, b] × [c, d] containing D and Define a new function G(x, y) on R by
(
f (x, y) if (x, y) is in D
G(x, y) =
0
if (x, y) is in R but not in D
Definition: We define the integral of f (x, y) over D to be
ZZ
ZZ
f (x, y)dA =
G(x, y)dA
D
R
provided the integral over the region R exists.
Note: G is likely to have discontinuities on the boundary of D. However, it can be shown that if f is
continuous on D and the boundary of D is ”well behaved”, then G is integrable over over R.
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Math 251-copyright Joe Kahlig, 15C
By Fubini’s Theorem we get:
ZZ
Zb
ZZ
f (x, y)dA =
G(x, y)dA =
D
Zd

G(x, y)dy  dx

a
R

c
Recall that G(x, y) = 0 if y < B(x) or if y > T (x). Therefor,
B(x)
Z
Zd
G(x, y)dy =
G(x, y)dy +
c
c
Zd
B(x)
Z
Zd
G(x, y)dy +
B(x)
c
c
Zd
T
Z(x)
G(x, y)dy =
Zd
f (x, y)dy +
B(x)
G(x, y)dy
T (x)
T
Z(x)
0dy +
G(x, y)dy =
c
T
Z(x)
0dy
T (x)
f (x, y)dy
B(x)
Thus
Theorem: If f is continuous on a type I region D such that
D = {(x, y)|a ≤ x ≤ b, B(x) ≤ y ≤ T (x)} then
ZZ
Zb TZ(x)
f (x, y)dA =
f (x, y)dydx
D
a B(x)
Definition: A plane region D is said to be of Type II if it lies between two continuous functions of
y, that is
D = {(x, y)|c ≤ y ≤ d, L(y) ≤ x ≤ R(y)}
Theorem: If f is continuous on a type II region D such that
D = {(x, y)|c ≤ y ≤ d, L(y) ≤ x ≤ R(y)}then
ZZ
Zd R(y)
Z
f (x, y)dA =
f (x, y)dxdy
D
c L(y)
Math 251-copyright Joe Kahlig, 15C
Example: If D is the region bounded by y = 4x and y = x2 evaluate
ZZ
x + y dA
D
Example: Evaluate the double integral of f (x, y) = x over the
region D = {(x, y)|0 ≤ x ≤ π, 0 ≤ y ≤ sin(x)}.
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Math 251-copyright Joe Kahlig, 15C
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Example: Set up the double integral that would evaluate the function f (x, y) = x + y over the region
bounded by the lines y = 6x, y = 2x, and y = −2x + 16
Example Find the volume of the solid bounded by the cylinder x2 + z 2 = 9, the planes x = 0, y = 0,
z = 0, x + 2y = 2 in the first octant.
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Math 251-copyright Joe Kahlig, 15C
Z1 Z2
Example: Evaluate the integral by changing the order of integration
cos(y 2 )dydx
0 2x
Properties of Double Integrals:
• If a and b are real numbers then
ZZ
ZZ
(af (x, y) + bg(x, y))dA = a
D
ZZ
f (x, y)dA + b
D
g(x, y)dA
D
• If D = D1 ∪ D2 , where D1 and D2 do not overlap except perhaps on their boundaries then
ZZ
ZZ
f (x, y)dA =
D
ZZ
f (x, y)dA +
D1
f (x, y)dA
D2
ZZ
• If we integrate f (x, y) = 1 over a region D, then
dA = the area of D.
D