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Math 2433
Section 26013
MW 1-2:30pm GAR 205
Bekki George
[email protected]
639 PGH
Office Hours:
11:00 - 11:45am MWF or by appointment
Popper14
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16.2 The Double Integral over a Rectangle
Definition: Let f = f(x, y) be continuous on the rectangle R: a < x < b, c < y < d.
Let P be a partition of R and let mij and Mij be the minimum and maximum values
of f on the i, j sub-rectangle Rij . Then
n
(i) Lower sum: Lf (P) =
m
∑ ∑ m Δx Δy
ij
i
j
i=1 j=1
n
m
∑ M ij Δxi Δy j
(ii) Upper sum: Uf (P) = ∑
i=1 j=1
Upper and Lower sums over a rectangle:
2
f
(x,
y)
=
x
− 2 y, 1 ≤ x ≤ 3, 0 ≤ y ≤ 4
Example: Find Lf (P) and Uf (P) on
P1 = {1, 2, 5/2, 3} and P2 = {0,1/2, 2, 4}
The double integral of f over R is the unique number I that satisfies
Lf (P) < I < Uf (P) for all partitions P.
Notation: I =
∫∫
ℜ
f (x, y) dx dy
Let Ω be an arbitrary closed bounded region in the plane. Then
∫∫
Ω
f ( x, y)dxdy = ∫∫ F ( x, y )dxdy
ℜ
where R is a rectangle that contains Ω, and F(x, y) = f(x, y) on Ω and F(x, y) = 0 on
R− Ω.
2
f
(x,
y)
=
x
+ 2 y, 1 ≤ x ≤ 2, 0 ≤ y ≤ 2 P1 = {1, 2} and
2. Find Lf (P) on
P2 = {0,1/2, 2}
a.
b.
c.
d.
e.
3
5/2
8
13
none of these
16.3 Repeated Integrals
If the region Ω is given by a < x < b, φ1 (x) ≤ y ≤ φ2 (x) (this is called a Type I
region), then
b φ2 ( x )
∫∫
Ω
f ( x, y )dxdy = ∫
a
∫
φ
f ( x, y )dydx
1 ( x)
If the region Ω is given by c < y < d, ψ 1 ( y) ≤ x ≤ ψ 2 ( y) (this is called a Type
II region), then
∫∫
Ω
d ψ2 ( y)
f ( x, y )dxdy = ∫
c
∫
ψ
1( y)
f ( x, y )dxdy
Applications of double integrals include
Volume: V =
Area: A =
∫∫
∫∫
Ω
Ω
f (x, y)dx dy, f (x, y) > 0, f (x, y) is top and Ω is base
dx dy, Ω is region to find area of
As well as Mass of a Plate and Center of Mass (later).
Examples:
1. Evaluate
3
x
∫∫ y dxdy
Ω
taking Ω : 0 ≤ x ≤ 1, 0 ≤ y ≤ x
2. Evaluate
∫∫ cos(x + y) dxdy
Ω
π
π
Ω
:
0
≤
x
≤
,
0
≤
y
≤
taking
2
2
3. Evaluate
4
2
(x
+
y
) dxdy
∫∫
Ω
y = x 3 and y = x 2
taking Ω is the region bounded between
4. Calculate by double integration the area bounded by the curves y = x and
x = 4y − y 2
2
2
z
=
x
+
y
5. Give the formula for the volume under the paraboloid
within the
cylinder
x 2 + y 2 ≤ 1, z ≥ 0
using double integrals.
2 4
2
2x
cos(y
)dy dx by changing the order of integration.
6. Calculate ∫ ∫2
0 x
7. Find the volume of the solid bounded by the coordinate planes and the plane
x y z
+ + =1
2 3 4
3. Evaluate the integral taking Ω : 0 < x < 1, 0 < y < 4.
4. Which of the following can be used to find the integral taking
Ω : 0 < x < 1, x2 < y < x.