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Multiple_Integrals.pdf
1. Multiple Integrals
Dr. Xuan Dieu Bui
School of Applied Mathematics and Informatics,
Hanoi University of Science and Technology
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 1 / 89
2. Bibliography
James Stewart, Calculus Early Transcendentals, Brooks Cole Cengage
Learning, 2012.
1 Multiple Integrals: Chapter 15,
2 Integrals depending on a parameter:
3 Line Integrals: Chapter 16,
4 Surface Integrals: Chapter 16,
5 Vector Calculus: Chapter 16,
6 Series: Chapter 11.
http://bit.ly/bai-giang
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 2 / 89
3. Multiple Integrals
1 Double Integrals
Double Integrals over Rectangles
Double Integrals over General Regions
Double Integrals in Polar Coordinates
Change of Variables in Double Integrals
Applications of Double Integrals
2 Triple Integrals
Triple Integrals over Rectangular Box
Triple Integrals over General Regions
Change of Variables in Triple Integrals
Triple Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
Applications of Triple Integrals
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 3 / 89
4. Double Integrals
Multiple Integals
1 Double Integrals
Double Integrals over Rectangles
Double Integrals over General Regions
Double Integrals in Polar Coordinates
Change of Variables in Double Integrals
Applications of Double Integrals
2 Triple Integrals
Triple Integrals over Rectangular Box
Triple Integrals over General Regions
Change of Variables in Triple Integrals
Triple Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
Applications of Triple Integrals
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 4 / 89
5. Double Integrals Double Integrals over Rectangles
Double Integrals
Review of the Definite Integral
Z b
a
f (x)dx = lim
n→∞
n
X
i=1
f (x∗
i )∆x.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 5 / 89
6. Double Integrals Double Integrals over Rectangles
Double Integrals
Review of the Definite Integral
1 divide [a, b] into n subintervals [xi−1, xi ] of equal width ∆x = b−a
n
2 choose sample points x∗
i in these subintervals,
3 form the Riemann sum
n
P
i=1
f (x∗
i )∆x
4 take the limit V =
b
R
a
f (x)dx = lim
n→∞
n
P
i=1
f (x∗
i )∆x
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 6 / 89
7. Double Integrals Double Integrals over Rectangles
Volumes and Double Integrals
Goal: find the volume of S := {(x, y, z) ∈ R3|0 ≤ z ≤ f (x, y), (x, y) ∈ R.}
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 7 / 89
8. Double Integrals Double Integrals over Rectangles
Volumes and Double Integrals
1 divide [a, b] into m subintervals and [c, d] into n subintervals, each of
equal width.
2 choose sample points (x∗
ij , y∗
ij ) ∈ Rij = [xi−1, xi ] × [yi−1, yi ],
V (Rij) ≈ f (x∗
ij , y∗
ij )∆x∆y.
3 Riemann Sum V =
m
P
i=1
n
P
j=1
Rij ≈
m
P
i=1
n
P
j=1
f (x∗
ij , y∗
ij )∆x∆y = Sm,n.
4 take the limit lim
m,n→∞
Smn.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 8 / 89
9. Double Integrals Double Integrals over Rectangles
Double Integrals
Definition
The double integral of f over the rectangle R is
ZZ
R
f (x, y)dxdy = lim
m,n→∞
m
X
i=1
n
X
j=1
f (x∗
ij , y∗
ij )∆x∆y
if this limit exists.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 9 / 89
10. Double Integrals Double Integrals over Rectangles
Fubini’s Theorem
Theorem (Fubini’s Theorem)
If f is continuous on the rectangle R = {(x, y)|a ≤ x ≤ b, c ≤ x ≤ d},
then
ZZ
R
f (x, y)dxdy =
Z b
a
Z d
c
f (x, y)dy
!
dx =
Z d
c
Z b
a
f (x, y)dx
!
dy.
RR
R
f (x, y)dxdy = V =
b
R
a
A(x)dx, where
A(x) =
d
R
c
f (x, y)dy.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 10 / 89
11. Double Integrals Double Integrals over Rectangles
Triple Integrals
Properties
1
RR
R
[f (x, y) + g (x, y)] dxdy =
RR
R
f (x, y) dxdy +
RR
R
g (x, y) dxdy.
2
RR
R
αf (x, y) dxdy = α
RR
R
f (x, y)dxdy.
3
RR
D
f (x, y) dxdy =
RR
D1
f (x, y) dxdy +
RR
D2
f (x, y) dxdy.
x
y
O
D1 D2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 11 / 89
12. Double Integrals Double Integrals over General Regions
Double Integrals over general regions
F(x, y) =
(
f (x, y), if (x, y) ∈ D,
0, if (x, y) ∈ R D.
and ZZ
D
f (x, y)dxdy =
ZZ
R
F(x, y)dxdy.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 12 / 89
13. Double Integrals Double Integrals over General Regions
Double Integrals over plane regions of type I
D = {(x, y)|a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}.
x
y
O
D
y = g1(x)
y = g2(x)
a b
R = [a, b] × [c, d]
c
d
By Fubini’s Theorem,
ZZ
D
f (x, y)dxdy =
ZZ
R
F(x, y)dxdy
14. Double Integrals Double Integrals over General Regions
Double Integrals over plane regions of type I
D = {(x, y)|a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}.
x
y
O
D
y = g1(x)
y = g2(x)
a b
R = [a, b] × [c, d]
c
d
By Fubini’s Theorem,
ZZ
D
f (x, y)dxdy =
ZZ
R
F(x, y)dxdy
15. Double Integrals Double Integrals over General Regions
Double Integrals over plane regions of type I
D = {(x, y)|a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}.
x
y
O
D
y = g1(x)
y = g2(x)
a b
R = [a, b] × [c, d]
c
d
By Fubini’s Theorem,
ZZ
D
f (x, y)dxdy =
ZZ
R
F(x, y)dxdy
=
Z b
a
dx
Z d
c
F(x, y)dy =
Z b
a
dx
Z g2(x)
g1(x)
f (x, y)dy.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 13 / 89
16. Double Integrals Double Integrals over General Regions
Double Integrals over general regions
Double Integrals over plane regions of type I
If f is continuous on a type I region D such that
D = {(x, y)|a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}
then ZZ
D
f (x, y)dxdy =
Z b
a
dx
Z g2(x)
g1(x)
f (x, y)dy.
x
y
O
D
a b
c
d
x
y
O
D
a b
c
d
x
y
O
D
a b
c
d
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 14 / 89
17. Double Integrals Double Integrals over General Regions
Double Integrals over general regions
Double Integrals over plane regions of type II
If f is continuous on a type II region D such that
D = {(x, y)|c ≤ y ≤ d, h1(y) ≤ x ≤ h2(y)}
then ZZ
D
f (x, y)dxdy =
Z d
c
dy
Z h2(y)
h1(y)
f (x, y)dx.
y
D
x = h1(y)
x = h2(y)
d
c
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 15 / 89
18. Double Integrals Double Integrals over General Regions
Example
Evaluate
RR
D
x2 (y − x) dxdy where D is the region bounded by y = x2 and
x = y2.
x
y
O 1
1
y = x2
x = y2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 16 / 89
19. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Example
Evaluate I =
RR
D
xey2
dx, where D = {(x, y) : 0 ≤ x ≤ 1, x2 ≤ y ≤ 1}.
x
1
y
1
O
y = x2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 17 / 89
20. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Exercise
Change the order of integration I =
b
R
a
dx
f2(x)
R
f1(x)
f (x, y)dy.
1 From the iterated integral, sketch the region of integration,
2 Divide it into regions of type II, for instance,
Di = {(x, y)|ci ≤ y ≤ di , gi (y) ≤ x ≤ hi (x)},
3
b
Z
a
dx
f2(x)
Z
f1(x)
f (x, y)dy =
X
i
di
Z
ci
dy
hi (y)
Z
gi (y)
f (x, y)dx.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 18 / 89
21. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Exercise
Change the order of integration I =
b
R
a
dx
f2(x)
R
f1(x)
f (x, y)dy.
1 From the iterated integral, sketch the region of integration,
2 Divide it into regions of type II, for instance,
Di = {(x, y)|ci ≤ y ≤ di , gi (y) ≤ x ≤ hi (x)},
3
b
Z
a
dx
f2(x)
Z
f1(x)
f (x, y)dy =
X
i
di
Z
ci
dy
hi (y)
Z
gi (y)
f (x, y)dx.
Similar for
d
R
c
dy
f2(y)
R
f1(y)
f (x, y)dx.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 18 / 89
22. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Example
Change the order of integration
1
R
−1
dx
1−x2
R
−
√
1−x2
f (x, y) dy.
x
1
y
1
O
D1
D2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 19 / 89
23. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Example
Change the order of integration
1
R
0
dy
1+
√
1−y2
R
2−y
f (x, y) dx.
x
2
1
y
2
1
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 20 / 89
24. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Example
Change the order of integration
2
R
0
dx
√
2x
R
√
2x−x2
f (x, y) dy.
x
2
1
y
2
1
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 21 / 89
25. Double Integrals Double Integrals over General Regions
Change The Order of Integration
Example
Change the order of integration
√
2
R
0
dy
y
R
0
f (x, y) dx+
2
R
√
2
dy
√
4−y2
R
0
f (x, y) dx.
x
√
2
y
√
2
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 22 / 89
26. Double Integrals Double Integrals over General Regions
Double Integrals involving Absolute Value Functions
Evaluate
RR
D
|f (x, y)| dxdy. The curve f (x, y) = 0 divides D into two
parts,
D+
= D ∩ {f (x, y) ≥ 0}, D−
= D ∩ {f (x, y) ≤ 0}.
ZZ
D
|f (x, y)| dxdy =
ZZ
D+
f (x, y) dxdy −
ZZ
D−
f (x, y) dxdy (1)
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 23 / 89
27. Double Integrals Double Integrals over General Regions
Double Integrals involving Absolute Value Functions
Evaluate
RR
D
|f (x, y)| dxdy. The curve f (x, y) = 0 divides D into two
parts,
D+
= D ∩ {f (x, y) ≥ 0}, D−
= D ∩ {f (x, y) ≤ 0}.
ZZ
D
|f (x, y)| dxdy =
ZZ
D+
f (x, y) dxdy −
ZZ
D−
f (x, y) dxdy (1)
Algorithm
1 Sketch the curve f (x, y) = 0 to find D+, D−.
2 Apply formula (1).
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 23 / 89
28. Double Integrals Double Integrals over General Regions
Double Integrals involving Absolute Value Functions
Example
Evaluate
RR
D
|x + y|dxdy, D :
(x, y) ∈ R2 ||x ≤ 1| , |y| ≤ 1 .
O
x
1
D+
y
1
D−
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 24 / 89
29. Double Integrals Double Integrals over General Regions
Double Integrals involving Absolute Value Functions
Example
Evaluate
RR
D
q
|y − x2|dxdy, D :
(x, y) ∈ R2 ||x| ≤ 1, 0 ≤ y ≤ 1 .
O x
1
D+
y
1
D−
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 25 / 89
30. Double Integrals Double Integrals over General Regions
Solving Double Integrals Using Symmetry
Theorem
If
1 f (x, y) is an odd function with respect to y and,
2 D is symmetric with respect to x− axis
then ZZ
D
f (x, y) dxdy = 0.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 26 / 89
31. Double Integrals Double Integrals over General Regions
Solving Double Integrals Using Symmetry
Theorem
If
1 f (x, y) is an odd function with respect to y and,
2 D is symmetric with respect to x− axis
then ZZ
D
f (x, y) dxdy = 0.
Theorem
If
1 f (x, y) is an even function with respect to y and,
2 D is symmetric with respect to x− axis
ZZ
D
f (x, y) dxdy = 2
ZZ
D+
f (x, y) dxdy.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 26 / 89
32. Double Integrals Double Integrals over General Regions
Solving Double Integrals Using Symmetry
Theorem
If
1 f (−x, −y) = −f (x, y),
2 D is symmetric with respect to the origin,
then
RR
D
f (x, y) dxdy = 0.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 27 / 89
33. Double Integrals Double Integrals over General Regions
Solving Double Integrals Using Symmetry
Theorem
If
1 f (−x, −y) = −f (x, y),
2 D is symmetric with respect to the origin,
then
RR
D
f (x, y) dxdy = 0.
Example
Evaluate
RR
|x|+|y|≤1
|x| + |y|dxdy.
O x
1
y
1
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 27 / 89
34. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
The polar coordinate of a point M is a pair (r, θ), where
r =
−
−
→
OM
θ =
−
−
→
OM,Ox.
O
y
x
O
M
θ
r = |
−
−
→
OM|
x
y
Polar coordinates vs rectangular coordinates:
(
x = r cos θ,
y = r sin θ.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 28 / 89
35. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
∆Ai =
1
2
r2
i ∆θ −
1
2
r2
i−1∆θ =
1
2
(r2
i − r2
i−1)∆θ
=
1
2
(ri + ri−1)(ri − ri−1)∆θ = r∗
i ∆r∆θ.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 29 / 89
36. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
ZZ
R
f (x, y)∆A = lim
m,n→∞
m
X
i=1
n
X
j=1
f (r∗
i cos θ∗
j , r∗
i sin θ∗
j )∆Ai
= lim
m,n→∞
m
X
i=1
n
X
j=1
f (r∗
i cos θ∗
j , r∗
i sin θ∗
j )r∗
i ∆r∆θ
=
ZZ
R
f (r cos θ, r sin θ) r drdθ.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 30 / 89
37. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
If f is continuous on a polar region of the form
(
θ1 ≤ θ ≤ θ2
r1 (θ) ≤ r ≤ r2 (θ) ,
then
I =
θ2
Z
θ1
dθ
r2(θ)
Z
r1(θ)
f (r cos θ, r sin θ) r dr
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 31 / 89
38. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
Example
Evaluate I =
RR
D
dxdy, where D :
(
4x ≤ x2 + y2 ≤ 8x,
x ≤ y ≤
√
3x.
x
4
2 8
y
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 32 / 89
39. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
Example
Evaluate
RR
D
dxdy
(x2+y2)2 , where D :
(
4y ≤ x2 + y2 ≤ 8y
x ≤ y ≤ x
√
3.
x
y
4
8
O
y = x
y = x
√
3
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 33 / 89
40. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
Example
Evaluate
RR
D
xy2dxdy where D is bounded by
(
x2 + (y − 1)2
= 1
x2 + y2 − 4y = 0.
x
y
2
4
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 34 / 89
41. Double Integrals Double Integrals in Polar Coordinates
Double Integrals in Polar Coordinates
Example
Evaluate
RR
D
xy
x2+y2 dxdy, where D :
(
x2 + y2 ≤ 12, x2 + y2 ≥ 2x
x2 + y2 ≥ 2
√
3y, x ≥ 0, y ≥ 0.
x
2 2
√
3
y
2
√
3
O
D1
D2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 35 / 89
42. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
Back in Calculus I,
b
R
a
f (x)dx =
d
R
c
f (x(t)) x′
(t) dt, where x = x(t).
Similarly,
RR
R
f (x, y)dxdy =
RR
S
f (x(u, v), y(u, v)) factor dudv
Example
Consider the transformation T :
(
x = x(u, v) = u2 − v2,
y = y(u, v) = 2uv.
Find the image of the square S = {(u, v) : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1}.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 36 / 89
43. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
∆A ≈ |ru × rv |∆u∆v =
x′
u x′
v
y′
u y′
v
∆u∆v,
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 37 / 89
44. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
∆A ≈ |ru × rv |∆u∆v =
x′
u x′
v
y′
u y′
v
∆u∆v, J =
D(x, y)
D(u, v)
=
x′
u x′
v
y′
u y′
v
.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 37 / 89
45. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
ZZ
R
f (x, y)dA = lim
m,n→∞
m
X
i=1
n
X
j=1
f (xi , yj)∆A
= lim
m,n→∞
m
X
i=1
n
X
j=1
f (x(ui , vj), y(ui , vj))
x′
u x′
v
y′
u y′
v
∆u∆v
=
ZZ
S
f (x(u, v), y(u, v))|J|dudv.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 38 / 89
46. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
Change of variables
Let T :
(
x = x(u, v),
y = y(u, v)
, S → R,
x(u, v), y(u, v) has continuous partial derivatives on S,
the transformation is a 1 − 1 map.
the Jacobian J =
x′
u x′
v
y′
u y′
v
6= 0 on S.
Then
RR
D
f (x, y)dxdy =
RR
S
f (x(u, v), y(u, v))|J|dudv.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 39 / 89
47. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
Change of variables
Let T :
(
x = x(u, v),
y = y(u, v)
, S → R,
x(u, v), y(u, v) has continuous partial derivatives on S,
the transformation is a 1 − 1 map.
the Jacobian J =
x′
u x′
v
y′
u y′
v
6= 0 on S.
Then
RR
D
f (x, y)dxdy =
RR
S
f (x(u, v), y(u, v))|J|dudv.
Example
Evaluate I =
RR
D
4x2 − 2y2
dxdy, where D :
(
1 ≤ xy ≤ 4
x ≤ y ≤ 4x.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 39 / 89
48. Double Integrals Change of Variables in Double Integrals
Change of variables in double integrals
Example
Evaluate I =
RR
D
4x2 − 2y2
dxdy, where D :
(
1 ≤ xy ≤ 4
x ≤ y ≤ 4x.
x
y
O 1
1
y = 4x
y = x
xy = 4
xy = 1
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 40 / 89
49. Double Integrals Change of Variables in Double Integrals
Polar coordinate transformation
1
(
x = r cos θ,
y = r sin θ
2 J = D(x,y)
D(r,θ) =
x′
r x′
θ,
y′
r y′
θ
= r
3
RR
R
f (x, y)dxdy =
RR
S
f (r cos θ, r sin θ) r drdθ.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 41 / 89
50. Double Integrals Change of Variables in Double Integrals
Polar coordinates
If D : x2
a2 + y2
b2 ≤ 1, then
(
x = ar cos ϕ
y = br sin ϕ
, J = abr
Example
Evaluate
RR
D
9x2 − 4y2 dxdy, where D : x2
4 + y2
9 ≤ 1.
x
2
y
3
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 42 / 89
51. Double Integrals Change of Variables in Double Integrals
Polar Coordinates
If D : (x − a)2
+ (y − b)2
≤ R2, then
(
x = a + r cos ϕ
y = b + r sin ϕ
, J = r
Example
Evaluate
2
R
0
dx
√
2x−x2
R
−
√
2x−x2
p
2x − x2 − y2dy.
x
2
y
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 43 / 89
52. Double Integrals Change of Variables in Double Integrals
Polar Coordinates
Example
Evaluate
RR
D
xydxdy, where D : (x − 2)2
+ y2 ≤ 1.
x
3
1
y
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 44 / 89
53. Double Integrals Change of Variables in Double Integrals
Polar Coordinates
Example
Evaluate
RR
D
xydxdy, where D : (x − 2)2
+ y2 ≤ 1, y ≥ 0
x
3
1
y
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 45 / 89
54. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain bounded by
(
y2 = x, y2 = 2x
x2 = y, x2 = 2y.
x
y
O
y = x2 x2 = 2y
x = y2
2x = y2
1 3
√
2 3
√
4 2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 46 / 89
55. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain D bounded by
(
y = 0, y2 = 4ax
x + y = 3a, (a 0) .
y
O x
3a
3a
−6a
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 47 / 89
56. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain D bounded by
(
x2 + y2 = 2x, x2 + y2 = 4x
x = y, y = 0.
y = x
x
2 4
y
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 48 / 89
57. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain D bounded by r = 1, r = 2
√
3
cos ϕ.
x
y
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 49 / 89
58. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain D bounded by
x2 + y2
2
= 2a2xy (a 0).
x
y
O
r = a
√
sin 2ϕ
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 50 / 89
59. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain D bounded by x3 + y3 = axy (a 0)
(Descartes leaf)
x
y
O 1
2
1
2
y = −x − 1
3
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 51 / 89
60. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the area of the domain D bounded by r = a (1 + cos ϕ) (a 0)
(Cardioids)
x
y
O
2a
a
−a
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 52 / 89
61. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Ω :
(
0 ≤ z ≤ f (x, y),
(x, y) ∈ D
⇒ V (Ω) =
ZZ
D
f (x, y) dxdy.
y
z
x
O
z = f (x, y)
D
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 53 / 89
62. Double Integrals Applications of Double Integrals
Volume of cylindrical objects
V :
(
z1(x, y) ≤ z ≤ z2(x, y),
(x, y) ∈ D
⇒ V (Ω) =
ZZ
D
(z2 (x, y) − z1 (x, y))dxdy.
y
z
x
O
z = z2(x, y)
z = z1(x, y)
D
Ω
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 54 / 89
63. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the volume of the object given by
(
3x + y ≥ 1, y ≥ 0
3x + 2y ≤ 2, 0 ≤ z ≤ 1 − x − y.
y
z
x
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 55 / 89
64. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the volume of the object bounded by the surfaces
(
z = 4 − x2 − y2
2z = 2 + x2 + y2.
y
z
x
O
2z = 2 + x2 + y2
z = 4 − x2 − y2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 56 / 89
65. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the volume of the object given by V :
(
0 ≤ z ≤ 1 − x2
− y2
y ≥ x, y ≤
√
3x
y
z
x
O 1
1
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 57 / 89
66. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the volume of the object given by V :
(
x2
+ y2
+ z2
≤ 4a2
x2
+ y2
− 2ay ≤ 0
y
z
x
O
2a
2a
2a
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 58 / 89
67. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the volume of the object bounded by the surfaces
z =
x2
a2
+
y2
b2
, z = 0
x2
a2
+
y2
b2
=
2x
a
x
z
O
z = x2
a2 + y2
b2
a
1
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 59 / 89
68. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Example
Compute the volume of the object bounded by the surfaces
V :
az = x2
+ y2
z =
q
x2 + y2
y
z
x
O a
−a
a
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 60 / 89
69. Double Integrals Applications of Double Integrals
Applications of Double Integrals
Area of a curved surface
S =
RR
D
q
1 + z′2
x + z′2
y dxdy.
y
z
x
O
z = f (x, y)
D
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 61 / 89
70. Triple Integrals
Multiple Integals
1 Double Integrals
Double Integrals over Rectangles
Double Integrals over General Regions
Double Integrals in Polar Coordinates
Change of Variables in Double Integrals
Applications of Double Integrals
2 Triple Integrals
Triple Integrals over Rectangular Box
Triple Integrals over General Regions
Change of Variables in Triple Integrals
Triple Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
Applications of Triple Integrals
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 62 / 89
71. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals
Review of the Definite Integral
Z b
a
f (x)dx = lim
n→∞
n
X
i=1
f (x∗
i )∆x.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 63 / 89
72. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals
Review of the Definite Integral
1 divide [a, b] into n subintervals [xi−1, xi ] of equal width ∆x = b−a
n
2 choose sample points x∗
i in these subintervals,
3 form the Riemann sum
n
P
i=1
f (x∗
i )∆x
4 take the limit V =
b
R
a
f (x)dx = lim
n→∞
n
P
i=1
f (x∗
i )∆x
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 64 / 89
73. Triple Integrals Triple Integrals over Rectangular Box
Volumes and Double Integrals
Goal: find the volume of S := {(x, y, z) ∈ R3|0 ≤ z ≤ f (x, y), (x, y) ∈ R.}
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 65 / 89
74. Triple Integrals Triple Integrals over Rectangular Box
Volumes and Double Integrals
1 divide [a, b] into m subintervals and [c, d] into n subintervals, each of
equal width.
2 choose sample points (x∗
ij , y∗
ij ) ∈ Rij = [xi−1, xi ] × [yi−1, yi ],
V (Rij) ≈ f (x∗
ij , y∗
ij )∆x∆y.
3 Riemann Sum V =
m
P
i=1
n
P
j=1
Rij ≈
m
P
i=1
n
P
j=1
f (x∗
ij , y∗
ij )∆x∆y = Sm,n.
4 take the limit lim
m,n→∞
Smn.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 66 / 89
75. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals over Rectangular Box
1 divide B into sub-boxes by
dividing [a, b] into l subintervals [xi−1, xi ] of equal width ∆x,
dividing [c, d] into m subintervals [yj−1, yj ] of equal width ∆y,
dividing [r, s] into n subintervals [zk−1, zk ] of equal width ∆z.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 67 / 89
76. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals over Rectangular Box
1 divide B into sub-boxes by
dividing [a, b] into l subintervals [xi−1, xi ] of equal width ∆x,
dividing [c, d] into m subintervals [yj−1, yj ] of equal width ∆y,
dividing [r, s] into n subintervals [zk−1, zk ] of equal width ∆z.
2 Choose sample points (x∗
ijk, y∗
ijk, z∗
ijk) in each box Bijk. Each sub-box
has volume ∆V = ∆x∆y∆z.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 67 / 89
77. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals over Rectangular Box
1 divide B into sub-boxes by
dividing [a, b] into l subintervals [xi−1, xi ] of equal width ∆x,
dividing [c, d] into m subintervals [yj−1, yj ] of equal width ∆y,
dividing [r, s] into n subintervals [zk−1, zk ] of equal width ∆z.
2 Choose sample points (x∗
ijk, y∗
ijk, z∗
ijk) in each box Bijk. Each sub-box
has volume ∆V = ∆x∆y∆z.
3 form the triple Riemann sum
l
P
i=1
m
P
j=1
n
P
k=1
f (x∗
ijk, y∗
ijk, z∗
ijk)∆V .
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 67 / 89
78. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals over Rectangular Box
Definition
ZZZ
B
f (x, y, z)dV = lim
l,m,n→∞
l
X
i=1
m
X
j=1
n
X
k=1
f (x∗
ijk, y∗
ijk, z∗
ijk)∆V .
Again, the triple integral always exists if is continuous.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 68 / 89
79. Triple Integrals Triple Integrals over Rectangular Box
Triple Integrals over Rectangular Box
Definition
ZZZ
B
f (x, y, z)dV = lim
l,m,n→∞
l
X
i=1
m
X
j=1
n
X
k=1
f (x∗
ijk, y∗
ijk, z∗
ijk)∆V .
Again, the triple integral always exists if is continuous.
Theorem (Fubini’s Theorem)
If f is continuous on the rectangular box B = [a, b] × [c, d] × [r, s] then
ZZZ
B
f (x, y, z)dxdydz =
Z b
a
dx
Z d
c
dy
Z s
r
f (x, y, z)dz.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 68 / 89
80. Triple Integrals Triple Integrals over General Regions
Triple Integrals over General Regions
If E is a general bounded solid region, choose
B = [a, b] × [c, d] × [r, s] ⊃ V
and define
F(x, y, z) =
(
f (x, y, z), if (x, y, z) ∈ E,
0, if (x, y, z) ∈ B E.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 69 / 89
81. Triple Integrals Triple Integrals over General Regions
Triple Integrals over General Regions
If E is a general bounded solid region, choose
B = [a, b] × [c, d] × [r, s] ⊃ V
and define
F(x, y, z) =
(
f (x, y, z), if (x, y, z) ∈ E,
0, if (x, y, z) ∈ B E.
Definition
ZZZ
E
f (x, y, z)dV =
ZZZ
B
F(x, y, z)dV .
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 69 / 89
82. Triple Integrals Triple Integrals over General Regions
Triple Integrals over Regions of type I
If V :
(
z1(x, y) ≤ z ≤ z2(x, y),
(x, y) ∈ D
then
I =
ZZZ
V
f (x, y, z) dxdydz =
ZZ
D
dxdy
z2(x,y)
Z
z1(x,y)
f (x, y, z) dz
y
z
x
O
z = z2(x, y)
z = z1(x, y)
D
V
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 70 / 89
83. Triple Integrals Triple Integrals over General Regions
Triple Integrals over Regions of type I
Reduce the triple integral into a double integral
1. Find the projection of V onto Oxy.
2. Find
(
the lower boundary z = z1 (x, y) ,
the upper boundary z = z2 (x, y)
of V .
3. Apply the formula
I =
ZZZ
V
f (x, y, z) dxdydz =
ZZ
D
dxdy
z2(x,y)
Z
z1(x,y)
f (x, y, z) dz
.
The idea:
Triple Integrals ⇒ Double Integrals ⇒ Iterated integrals
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 71 / 89
84. Triple Integrals Triple Integrals over General Regions
Triple Integrals over General Regions
Example
Evaluate
RRR
V
x2 + y2
dxdydz, where V :
p
x2 + y2 ≤ z ≤
p
1 − x2 − y2.
y
z
x
O
z =
p
1 − x2 − y2
z =
p
x2 + y2
D
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 72 / 89
85. Triple Integrals Triple Integrals over General Regions
Triple Integrals over General Regions
By the same sort of argument,
1 Type II:
RRR
E
f (x, y, z)dxdydz =
RR
D
u2(y,z)
R
u1(y,z)
f (x, y, z)dx
#
dydz.
2 Type III:
RRR
E
f (x, y, z)dxdydz =
RR
D
u2(x,z)
R
u1(x,z)
f (x, y, z)dy
#
dxdz.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 73 / 89
86. Triple Integrals Triple Integrals over General Regions
Solving triple integrals using symmetry
Theorem
If
1 V is symmetric with respect to z = 0,
2 f (x, y, z) is an odd function with respect to z
then
RRR
V
f (x, y, z) dxdydz = 0.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 74 / 89
87. Triple Integrals Triple Integrals over General Regions
Solving triple integrals using symmetry
Theorem
If
1 V is symmetric with respect to z = 0,
2 f (x, y, z) is an odd function with respect to z
then
RRR
V
f (x, y, z) dxdydz = 0.
Theorem
If
1 V is symmetric with respect to z = 0,
2 f (x, y, z) is an even function with respect to z
then
RRR
V
f (x, y, z) dxdydz = 2
RRR
V +
f (x, y, z) dxdydz.
Note: The role of x, y, z can be interchangeable.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 74 / 89
88. Triple Integrals Change of Variables in Triple Integrals
Change of Variables in Triple Integrals
1 Calculus 1,
b
R
a
f (x)dx =
d
R
c
f (x(u)) x′
(u) du, where x = x(u).
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 75 / 89
89. Triple Integrals Change of Variables in Triple Integrals
Change of Variables in Triple Integrals
1 Calculus 1,
b
R
a
f (x)dx =
d
R
c
f (x(u)) x′
(u) du, where x = x(u).
2
RR
R
f (x, y)dxdy =
RR
S
f (x(u, v), y(u, v))|J|dudv,
(
x = x(u, v),
y = y(u, v).
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 75 / 89
90. Triple Integrals Change of Variables in Triple Integrals
Change of Variables in Triple Integrals
1 Calculus 1,
b
R
a
f (x)dx =
d
R
c
f (x(u)) x′
(u) du, where x = x(u).
2
RR
R
f (x, y)dxdy =
RR
S
f (x(u, v), y(u, v))|J|dudv,
(
x = x(u, v),
y = y(u, v).
3
ZZZ
B
f (x, y, z)dxdydz =
ZZ
S
f (x(u, v, w), y(u, v, w), z(u, v, w)) factor dudvdw,
where
x = x(u, v, w),
y = y(u, v, w),
z = z(u, v, w).
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 75 / 89
91. Triple Integrals Change of Variables in Triple Integrals
Change of Variables in Triple Integrals
Consider the transformation: T :
x = x(u, v, w)
y = y(u, v, w)
z = z(u, v, w).
, V ′ → V satisfies
1 T is a 1 − 1 map.
2 x(u, v, w), y(u, v, w), z(u, v, w) are continuous and have continuous
partial derivatives on V ′.
3 The Jacobian determinant
J =
D(x, y, z)
D(u, v, w)
=
x′
u x′
v x′
w
y′
u y′
v y′
w
z′
u z′
v z′
w
6= 0 in V ′
.
Then
ZZZ
V
f (x, y, z)dxdydz =
ZZZ
V ′
f (x(u, v, w), y(u, v, w), z(u, v, w)) |J|dudvdw.
(1)
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 76 / 89
92. Triple Integrals Change of Variables in Triple Integrals
Change of Variables in Triple Integrals
Example
Evaluate
RRR
V
(x + y + z)dxdydz, where V is bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2
.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 77 / 89
93. Triple Integrals Change of Variables in Triple Integrals
Change of Variables in Triple Integrals
Example
Evaluate
RRR
V
(x + y + z)dxdydz, where V is bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2
.
Consider the transformation
u = x + y + z
v = x + 2y − z
w = x + 4y + z
J−1
=
D (u, v, w)
D (x, y, z)
=
1 1 1
1 2 −1
1 4 1
= 6 ⇒ J =
1
6
.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 77 / 89
94. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
z
y
x
O
M
M′
ϕ
r = |
−
−
→
OM′|
b
b
Cylindrical Coordinate
M(r, ϕ, z) , where (r, ϕ) is the
polar coordinate of M′.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 78 / 89
95. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
z
y
x
O
M
M′
ϕ
r = |
−
−
→
OM′|
b
b
Cylindrical Coordinate
M(r, ϕ, z) , where (r, ϕ) is the
polar coordinate of M′.
Cylindrical vs rectangular coor.
x = r cos ϕ, y = r sin ϕ, z = z.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 78 / 89
96. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
z
y
x
O
M
M′
ϕ
r = |
−
−
→
OM′|
b
b
Cylindrical Coordinate
M(r, ϕ, z) , where (r, ϕ) is the
polar coordinate of M′.
Cylindrical vs rectangular coor.
x = r cos ϕ, y = r sin ϕ, z = z.
The transformation
Taking the transformation
x = r cos ϕ
y = r sin ϕ
z = z.
then J = D(x,y,z)
D(r,ϕ,z) = r and
I =
ZZZ
V
f (x, y, z)dxdydz =
ZZZ
Vrϕz
f (r cos ϕ, r sin ϕ, z)rdrdϕdz.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 78 / 89
97. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
Example
Evaluate
RRR
V
x2 + y2
dxdydz, where V :
(
x2
+ y2
≤ 1
1 ≤ z ≤ 2
y
z
x
O
V
1
2
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 79 / 89
98. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
Example
Evaluate
RRR
V
z
p
x2 + y2dxdydz, where:
a) V is bounded by: x2 + y2 = 2x and z = 0, z = a (a 0).
b) V is a half of the sphere x2 + y2 + z2 ≤ a2, z ≥ 0 (a 0)
y
z
x
O
y
z
x
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 80 / 89
99. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
Example
Evaluate I =
RRR
V
ydxdydz, where V is bounded by:
(
y =
p
z2 + x2
y = h.
y
z
x
O h
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 81 / 89
100. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
Example
Evaluate I =
RRR
V
p
x2 + y2dxdydz where V is bounded by:
(
x2
+ y2
= z2
z = 1.
y
z
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 82 / 89
101. Triple Integrals Triple Integrals in Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
Example
Evaluate
RRR
V
dxdydz
√
x2+y2+(z−2)2
, where V :
(
x2
+ y2
≤ 1
|z| ≤ 1.
y
z
x
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 83 / 89
102. Triple Integrals Triple Integrals in Spherical Coordinates
Triple Integrals in Spherical Coordinates
z
y
x
O
M
M′
ϕ
θ
r = |
−
−
→
OM|
b
b
The spherical coordinate of a point M is an
ordered triple (r, θ, ϕ), where
r =
−
−
→
OM , θ =
Oz,
−
−
→
OM
, ϕ =
Ox,
−
−
→
OM′
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 84 / 89
103. Triple Integrals Triple Integrals in Spherical Coordinates
Triple Integrals in Spherical Coordinates
z
y
x
O
M
M′
ϕ
θ
r = |
−
−
→
OM|
b
b
The spherical coordinate of a point M is an
ordered triple (r, θ, ϕ), where
r =
−
−
→
OM , θ =
Oz,
−
−
→
OM
, ϕ =
Ox,
−
−
→
OM′
Spherical vs rectangular coor.
x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 84 / 89
104. Triple Integrals Triple Integrals in Spherical Coordinates
Triple Integrals in Spherical Coordinates
z
y
x
O
M
M′
ϕ
θ
r = |
−
−
→
OM|
b
b
The spherical coordinate of a point M is an
ordered triple (r, θ, ϕ), where
r =
−
−
→
OM , θ =
Oz,
−
−
→
OM
, ϕ =
Ox,
−
−
→
OM′
Spherical vs rectangular coor.
x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ.
The transformation
Let
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ,
then J = D(x,y,z)
D(r,θ,ϕ) = −r2 sin θ and
ZZZ
V
f (x, y, z)dxdydz =
ZZZ
Vrθϕ
f (· · · , · · · , · · · )r2
sin θdrdθdϕ.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 84 / 89
105. Triple Integrals Triple Integrals in Spherical Coordinates
Triple Integrals in Spherical Coordinates
Example
Evaluate
RRR
V
x2 + y2 + z2
dxdydz, where V :
(
1 ≤ x2
+ y2
+ z2
≤ 4
x2
+ y2
≤ z2
.
y
z
x
O
V1
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 85 / 89
106. Triple Integrals Triple Integrals in Spherical Coordinates
Triple Integrals in Spherical Coordinates
Example
Evaluate
RRR
V
p
x2 + y2 + z2dxdydz, where V : x2 + y2 + z2 ≤ z.
y
z
x
O
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 86 / 89
107. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
1 V : x2
a2 + y2
b2 + z2
c2 ≤ 1 ⇒
x = ar sin θ cos ϕ
y = br sin θ sin ϕ
z = cr cos θ
, J = −abcr2 sin θ
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 87 / 89
108. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
1 V : x2
a2 + y2
b2 + z2
c2 ≤ 1 ⇒
x = ar sin θ cos ϕ
y = br sin θ sin ϕ
z = cr cos θ
, J = −abcr2 sin θ
2 V : (x − a)2
+ (y − b)2
+ (z − c)2
≤ R2
⇒
x = a + r sin θ cos ϕ
y = b + r sin θ sin ϕ
z = c + r cos θ
, J = −r2
sin θ
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 87 / 89
109. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
1 V : x2
a2 + y2
b2 + z2
c2 ≤ 1 ⇒
x = ar sin θ cos ϕ
y = br sin θ sin ϕ
z = cr cos θ
, J = −abcr2 sin θ
2 V : (x − a)2
+ (y − b)2
+ (z − c)2
≤ R2
⇒
x = a + r sin θ cos ϕ
y = b + r sin θ sin ϕ
z = c + r cos θ
, J = −r2
sin θ
3 V : x2+y2
a2 + z2
b2 ≤ 1 ⇒
z = bz′
x = ar cos ϕ
y = ar sin ϕ
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 87 / 89
110. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
Example
Evaluate
RRR
V
z
p
x2 + y2dxdydz, where V : x2+y2
a2 + z2
b2 ≤ 1, z ≥ 0.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 88 / 89
111. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
Example
Evaluate
RRR
V
z
p
x2 + y2dxdydz, where V : x2+y2
a2 + z2
b2 ≤ 1, z ≥ 0.
Cylindrical Coordinate.
Let
z = bz′
x = ar cos ϕ
y = ar sin ϕ
, then
Spherical Coordinate.
Let
x = ar sin θ cos ϕ
y = ar sin θ sin ϕ
z = br cos θ
, then
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 88 / 89
112. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
Example
Evaluate
RRR
V
z
p
x2 + y2dxdydz, where V : x2+y2
a2 + z2
b2 ≤ 1, z ≥ 0.
Cylindrical Coordinate.
Let
z = bz′
x = ar cos ϕ
y = ar sin ϕ
, then
Spherical Coordinate.
Let
x = ar sin θ cos ϕ
y = ar sin θ sin ϕ
z = br cos θ
, then
J = a2br and
0 ≤ ϕ ≤ 2π,
0 ≤ r ≤ 1,
0 ≤ z′ ≤
√
1 − r2.
J = a2br2 sin θ and
0 ≤ ϕ ≤ 2π,
0 ≤ θ ≤ π
2 ,
0 ≤ r ≤ 1.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 88 / 89
113. Triple Integrals Triple Integrals in Spherical Coordinates
Spherical and Cylindrical Coordinates - Special Cases
Example
Evaluate
RRR
V
z
p
x2 + y2dxdydz, where V : x2+y2
a2 + z2
b2 ≤ 1, z ≥ 0.
Cylindrical Coordinate.
Let
z = bz′
x = ar cos ϕ
y = ar sin ϕ
, then
Spherical Coordinate.
Let
x = ar sin θ cos ϕ
y = ar sin θ sin ϕ
z = br cos θ
, then
J = a2br and
0 ≤ ϕ ≤ 2π,
0 ≤ r ≤ 1,
0 ≤ z′ ≤
√
1 − r2.
J = a2br2 sin θ and
0 ≤ ϕ ≤ 2π,
0 ≤ θ ≤ π
2 ,
0 ≤ r ≤ 1.
I =
2πa3b2
15
.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 88 / 89
114. Triple Integrals Applications of Triple Integrals
Volume of a solid Object
V =
ZZZ
V
dxdydz.
Example
Compute the volume of the domain V bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 89 / 89
115. Triple Integrals Applications of Triple Integrals
Volume of a solid Object
V =
ZZZ
V
dxdydz.
Example
Compute the volume of the domain V bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2.
Change of variables
u = x + y + z
v = x + 2y − z
w = x + 4y + z
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 89 / 89
116. Triple Integrals Applications of Triple Integrals
Volume of a solid Object
V =
ZZZ
V
dxdydz.
Example
Compute the volume of the domain V bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2.
Change of variables
u = x + y + z
v = x + 2y − z
w = x + 4y + z
J−1
=
D (u, v, w)
D (x, y, z)
=
1 1 1
1 2 −1
1 4 1
= 6
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 89 / 89
117. Triple Integrals Applications of Triple Integrals
Volume of a solid Object
V =
ZZZ
V
dxdydz.
Example
Compute the volume of the domain V bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2.
Change of variables
u = x + y + z
v = x + 2y − z
w = x + 4y + z
J−1
=
D (u, v, w)
D (x, y, z)
=
1 1 1
1 2 −1
1 4 1
= 6 ⇒ J =
1
6
,
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 89 / 89
118. Triple Integrals Applications of Triple Integrals
Volume of a solid Object
V =
ZZZ
V
dxdydz.
Example
Compute the volume of the domain V bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2.
Change of variables
u = x + y + z
v = x + 2y − z
w = x + 4y + z
J−1
=
D (u, v, w)
D (x, y, z)
=
1 1 1
1 2 −1
1 4 1
= 6 ⇒ J =
1
6
, V =
1
6
ZZZ
Vuvw
dudvdw = 8.
Dr. Xuan Dieu Bui Multiple Integrals I ♥ HUST 89 / 89