13.3 Double Integrals over General Regions
In this section, we study how to integrate a function not just over rectangles but also
over regions of more general shape.
Let D be a bounded region enclosed in a rectangular region R and f (x, y) a continuous function over D. We dene
(
F (x, y) =
f (x, y)
0
if (x, y) is in D
if (x, y) is in R but not in D
If F is integrable over R, then we dene the double integral of f over D by
¨
¨
f (x, y) dA :=
D
F (x, y) dA
R
: If f (x, y) ≥ 0 and f is continuous on the region D then the volume V of the
solid S that lies above D and under the graph of f is
Fact
¨
V =
f (x, y) dA
D
Example 1. Evaluate the double integral
˜p
D
16 − x2 − y 2 dA by identifying it as a
volume of a solid, where D = { (x, y) ∈ R2 | x2 + y 2 ≤ 16 }.
:
Properties of double integrals
1. If α and β are real numbers, then
D
D
D
2. If f (x, y) ≥ g(x, y), then
¨
¨
f (x, y) dA ≥
3.
g(x, y) dA
D
1 dA = the area of D
dA =
D
D
¨
g(x, y) dA
f (x, y) dA + β
(αf (x, y) + βg(x, y)) dA = α
¨
¨
¨
¨
D
4. If D = D1 ∪D2 , where D1 and D2 do not overlap except perhaps on their boundaries,
then
¨
¨
¨
f (x, y) dA =
D
f (x, y) dA +
D1
f (x, y) dA
D2
Example 2. If D = { (x, y) ∈ R2 | x2 + y 2 ≤ 16 }, then
˜
D
dA =
A plane region D is said to be of
continuous functions of x, that is,
type I
if it lies between the graphs of two
D = {(x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)}
where g1 and g2 are continuous on [a, b].
. If D is a region of type I such that D = {(x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤
g2 (x)}, then
Theorem
ˆ bˆ
¨
g2 (x)
f (x, y) dydx
f (x, y) dA =
D
Example 3. Evaluate the double integral
a
˜
D
g1 (x)
(x + y) dA where D is the region bounded
by the lines x = 2, y = x and the hyperbola xy = 1.
A plane region D is said to be of
continuous functions of y , that is,
type II
if it lies between the graphs of two
D = {(x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y)}
where h1 and h2 are continuous on [c, d].
. If D is a region of type II such that D = {(x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤
h2 (x)}, then
Theorem
¨
ˆ
d ˆ h2 (y)
f (x, y) dxdy
f (x, y) dA =
D
Example 4. Evaluate the double integral
with vertices (0, 0), (2, 4) and (6, 0).
h1 (y)
c
˜
D
xy dA where D is the triangular region
Example 5. Find the volume of the solid bounded by the cylinder x2 + y 2 = 1 and the
planes x = 0, y = z , z = 0 in the rst octant.
Example 6. Evaluate the integral by reversing the order of integration:
ˆ 1ˆ
1
x3 sin(y 3 ) dydx
0
x2
Example 7. Sketch the region of integration and change the order of integration:
ˆ 1ˆ
√
ˆ 2ˆ
x
f (x, y) dydx
f (x, y) dydx +
0
0
2−x
1
0