double integrals introduction. volume and double integral z=f(x,y) ≥ 0 on rectangle r=[a,b]×[c,d]...
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![Page 1: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/1.jpg)
Double Integrals
Introduction
![Page 2: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/2.jpg)
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 = ?
![Page 3: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/3.jpg)
![Page 4: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/4.jpg)
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
![Page 5: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/5.jpg)
m
i
n
jijij AyxfV
1 1
** ),(
Definition
m
i
n
jijij AyxfV
1 1
**
nm,
),(lim
![Page 6: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/6.jpg)
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
** ),(
![Page 7: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/7.jpg)
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
![Page 8: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/8.jpg)
Example 1
z=16-x2-2y2
0≤x≤20≤y≤2
Estimate the volume of the solid above the square and below the graph
![Page 9: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/9.jpg)
m=n=4 m=n=8 m=n=16V≈41.5 V≈44.875 V≈46.46875
Exact volume? V=48
![Page 10: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/10.jpg)
Example 2
z
?1
]2,2[]1,1[
2
R
dAx
R
![Page 11: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/11.jpg)
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 ),(),(
![Page 12: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/12.jpg)
Properties
AAA
dAyxgdAyxfdAyxgyxf ),(),()],(),([
AA
dAyxfcdAyxcf ),(),(
AA
dAyxgdAyxf ),(),(
Linearity
If f(x,y)≥g(x,y) for all (x,y) in R, then
Comparison
![Page 13: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/13.jpg)
2121
),(),(),(AAAA
dAyxfdAyxfdAyxf
Additivity
If A1 and A2 are non-overlapping regions then
Area
AdAdAAA
of area1
A1A2
![Page 14: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/14.jpg)
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
![Page 15: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/15.jpg)
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
![Page 16: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/16.jpg)
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
![Page 17: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/17.jpg)
Note
If f (x, y) = φ (x) ψ(y) then
d
c
b
a
d
c
b
aR
dyydxxdxdyyxdAyxf )()()()(),(
![Page 18: Double Integrals Introduction. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R 3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume](https://reader033.vdocuments.net/reader033/viewer/2022061413/56649f1b5503460f94c30a3a/html5/thumbnails/18.jpg)
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)