Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]...
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Transcript of Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]...
Double Integrals
Introduction
Volume and Double Integral
z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]
S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R}
Volume of S = ?
Volume of ij’s column: Ayxf ijij ),( **
m
i
n
jijij Ayxf
1 1
** ),(Total volume of all columns:
ij’s column:
Area of Rij is Δ A = Δ x Δ y
f (xij*, yij
*)
Δ y Δ xxy
z
Rij
(xi, yj)
Sample point (xij*, yij
*)x
y
m
i
n
jijij AyxfV
1 1
** ),(
Definition
m
i
n
jijij AyxfV
1 1
**
nm,
),(lim
Definition:
The double integral of f over the rectangle R is
if the limit exists
R
dAyxf ),(
m
i
n
jijij
R
AyxfdAyxf1 1
**
nm,
),(),( lim
Double Riemann sum:
m
i
n
jijij Ayxf
1 1
** ),(
Note 1. If f is continuous then the limit exists and the integral is defined
Note 2. The definition of double integral does not depend on the choice of sample points
If the sample points are upper right-hand corners then
m
i
n
jji
R
AyxfdAyxf1 1nm,
),(),( lim
Example 1
z=16-x2-2y2
0≤x≤20≤y≤2
Estimate the volume of the solid above the square and below the graph
m=n=4 m=n=8 m=n=16V≈41.5 V≈44.875 V≈46.46875
Exact volume? V=48
Example 2
z
?1
]2,2[]1,1[
2
R
dAx
R
Integrals over arbitrary regions
A
R
f (x,y)
0
• A is a bounded plane region
• f (x,y) is defined on A• Find a rectangle R
containing A• Define new function on R:
otherwise ,0
),( if ),(),(
Ayxyxfyxf
RA
dAyxfdAyxf ),(),(
Properties
AAA
dAyxgdAyxfdAyxgyxf ),(),()],(),([
AA
dAyxfcdAyxcf ),(),(
AA
dAyxgdAyxf ),(),(
Linearity
If f(x,y)≥g(x,y) for all (x,y) in R, then
Comparison
2121
),(),(),(AAAA
dAyxfdAyxfdAyxf
Additivity
If A1 and A2 are non-overlapping regions then
Area
AdAdAAA
of area1
A1A2
Computation• If f (x,y) is continuous on rectangle R=[a,b]×[c,d]
then double integral is equal to iterated integral
a bx
y
c
d
x
y
b
a
d
c
d
c
b
aR
dydxyxfdxdyyxfdAyxf ),(),(),(
fixed fixed
More general case• If f (x,y) is continuous on
A={(x,y) | x in [a,b] and h (x) ≤ y ≤ g (x)} then double integral is equal to iterated integral
a bx
y
h(x)
g(x)
x
b
a
xg
xhA
dydxyxfdAyxf)(
)(
),(),(
A
Similarly• If f (x,y) is continuous on
A={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral
d
x
y
d
c
yg
yhR
dxdyyxfdAyxf)(
)(
),(),(
c
h(y) g(y)y
A
Note
If f (x, y) = φ (x) ψ(y) then
d
c
b
a
d
c
b
aR
dyydxxdxdyyxdAyxf )()()()(),(
Examples
],2/[]1,2/1[ ,)sin( AdAxyyR
2
A
x dAe where A is a triangle with vertices(0,0), (1,0) and (1,1)