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1. Double Integrals
Introduction

2. 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 = ?

4. Volume of ij’s column: Ayxf ijij ∆),( **
∑∑= =
∆
m
i
n
j
ijij 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
x
x
y
z
Rij
(xi, yj)
Sample point (xij
*
, yij
*
)
x
y

5. ∑∑= =
∆≈
m
i
n
j
ijij AyxfV
1 1
**
),(
Definition
∑∑= =∞→
∆=
m
i
n
j
ijij AyxfV
1 1
**
nm,
),(lim

6. Definition:
The double integral
of f over the rectangle R is
if the limit exists
∫∫R
dAyxf ),(
∑∑∫∫ = =∞→
∆=
m
i
n
j
ijij
R
AyxfdAyxf
1 1
**
nm,
),(),( lim
Double Riemann sum: ∑∑= =
∆
m
i
n
j
ijij Ayxf
1 1
**
),(

7. 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
j
ji
R
AyxfdAyxf
1 1nm,
),(),( lim

8. Example 1
z=16-x2
-2y2
0≤x≤2
0≤y≤2
Estimate the volume
of the solid above
the square and
below the graph

9. m=n=4 m=n=8 m=n=16
V≈41.5 V≈44.875 V≈46.46875
Exact volume? V=48

10. Example 2
z
?1
]2,2[]1,1[
2
=−
−×−=
∫∫R
dAx
R

11. 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),(
),(
Ayxyxf
yxf
∫∫∫∫ =
RA
dAyxfdAyxf ),(),(

12. Properties
∫∫∫∫∫∫ +=+
AAA
dAyxgdAyxfdAyxgyxf ),(),()],(),([
∫∫∫∫ =
AA
dAyxfcdAyxcf ),(),(
∫∫∫∫ ≥
AA
dAyxgdAyxf ),(),(
Linearity
If f(x,y)≥g(x,y) for all (x,y) in R, then
Comparison

13. ∫∫∫∫∫∫ +=
∪ 2121
),(),(),(
AAAA
dAyxfdAyxfdAyxf
Additivity
If A1 and A2 are non-overlapping regions then
Area
AdAdA
AA
ofarea1 == ∫∫∫∫
A1
A2

14. Computation
• If f (x,y) is continuous on rectangle R=[a,b]×[c,d]
then double integral is equal to iterated integral
a b
x
y
c
d
x
y
∫∫∫∫∫∫ ==
b
a
d
c
d
c
b
aR
dydxyxfdxdyyxfdAyxf ),(),(),(
fixed fixed

15. 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 b
x
y
h(x)
g(x)
x
∫ ∫∫∫ =
b
a
xg
xhA
dydxyxfdAyxf
)(
)(
),(),(
A

16. 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

17. Note
If f (x, y) = φ (x) ψ(y) then






⋅





== ∫∫∫∫∫∫
d
c
b
a
d
c
b
aR
dyydxxdxdyyxdAyxf )()()()(),( ψϕψϕ

18. Examples
],2/[]1,2/1[,)sin( ππ×=∫∫ AdAxyy
R
2
∫∫
−
A
x
dAe where A is a triangle with vertices
(0,0), (1,0) and (1,1)