Triple integrals in cylindrical and spherical coordinates
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
MATH 209—Calculus, III
Volker Runde
University of Alberta
Edmonton, Fall 2011
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Cylindrical coordinates
Rectangular versus cylindrical coordinates
x = r cos θ,
y = r sin θ,
z = z,
r 2 = x 2 + y 2 .
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in cyclindrical coordinations, I
The setup
Let E ⊂ R 3 be of type I, i.e.,
with
E = {(x, y, z) : (x, y) ∈ D, u1(x, y) ≤ z ≤ u2(x, y)},
D = {(r, θ) : α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}
in polar coordinates.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in cyclindrical coordinations, II
Theorem (change to cylindrical coordinates in a tiple integral)
Given E and D as before:
���
f (x, y, z) dV
E
�� �� �
u2(x,y)
=
f (x, y, z) dz dA
D
� β � h2(θ)
=
α
h1(θ)
u1(x,y)
� u2(r cos θ,r sin θ)
u1(r cos θ,r sin θ)
f (r cos θ, r sin θ, z)r dz dr θ.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, I
Example
A solid E lies inside the cylinder x 2 + y 2 = 1, below the plane
z = 4, and above the paraboloid z = 1 − x 2 − y 2 .
Its density ρ(x, y, z) at any point is proportional to the point’s
distance from the z-axis.
What is the mass of E?
We have
and
E = {(x, y, z) : x 2 + y 2 ≤ 1, 1 − x 2 − y 2 ≤ z ≤ 4}
= {(r, θ, z) : 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1, 1 − r 2 ≤ z ≤ 4}
ρ(x, y, z) = C � x 2 + y 2 = Cr.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, II
Example (continued)
Thus:
���
m = C
= C
E
� 2π � 1
0
0
� x 2 + y 2 dV = C
� 2π � 1 � 4
r 2 (4 − (1 − r 2 )) dr dθ = 2Cπ
�
= 2Cπ r 3 �
r 5 �
+ �
5 �
0
0
1−r 2
� 1
0
r=1
r=0
r 2 dz dr dθ
3r 2 + r 4 dr
�
= 12Cπ
.
5
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Spherical coordinates coordinates
Rectangular versus spherical coordinates
x = ρ sin φ cos θ,
y = ρ sin φ sin θ,
z = ρ cos φ,
ρ 2 = x 2 + y 2 + z 2 .
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in spherical coordinates, I
Definition
A set of the form
{(r, θ, φ) : r ∈ [a, b], θ ∈ [α, β], φ ∈ [c, d]}
with a ≥ 0, β − α ≤ 2π, and d − c ≤ π is called a spherical
wedge.
Question
How does one evaluate ���
wedge?
E
f (x, y, z) dV if E is a spherical
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in spherical coordinates, II
As in rectangular coordinates. . .
Divide E into small spherical wedges Ej,k,ℓ, and pick a support
point (x ∗ j,k,ℓ , y ∗
j,k,ℓ , z∗ j,k,ℓ ) ∈ Ej,k,ℓ.
Then:
���
f (x, y, z) dV
with
E
= lim
n,m,ν→∞
n�
m�
ν�
j=1 k=1 ℓ=1
∆Vj,k,ℓ = volume of Ej,k,ℓ.
f (x ∗ ∗
j,k,ℓ , yj,k,ℓ , z∗ j,k,ℓ )∆Vj,k,ℓ
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in spherical coordinates, III
Question
What is the volume of Ej,k,ℓ, i.e., a very small spherical wedge?
Answer
∆Vj,k,ℓ ≈ ρ 2 j sin φℓ∆ρ∆θ∆φ.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in spherical coordinates, IV
Consequence
���
E
f (x, y, z) dV = lim
n,m,ν→∞
n�
m�
ν�
j=1 k=1 ℓ=1
f (ρ ∗ j sin φ ∗ ℓ cos θ∗ k , ρ∗ j sin φ ∗ ℓ sin θ∗ k , ρ∗ j cos φ ∗ ℓ )ρ2 sin φ∆ρ∆θ∆φ
� d � β � b
=
f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)
c
α
a
ρ 2 sin φ dρ dθ dφ.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Integration in spherical coordinates, V
Theorem (change to spherical coordinates in a tiple integral)
Let
E := {(ρ, θ, φ) : α ≤ θ ≤ β,
Then:
���
E
f (x, y, z) dV
c ≤ φ ≤ d, ≤ g1(θ, φ) ≤ ρ ≤ g2(θ, φ)}.
� d � β � g2(θ,φ)
=
f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)
c
α
g1(θ,φ)
ρ 2 sin φ dρ dθ dφ.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, III
Example
Let
Evaluate ���
In spherical coordinates:
E = {(x, y, z) : x 2 + y 2 + z 2 ≤ 1}.
E
e (x2 +y 2 +z 2 ) 3 2 dV .
E = {(ρ, θ, φ) : ρ ∈ [0, 1], θ ∈ [0, 2π], φ ∈ [0, π]}.
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, IV
Example (continued)
Then:
���
E
e (x2 +y 2 +z 2 ) 3 2 dV
� π
=
0
� π
=
0
� 2π � 1
e
0 0
(ρ2 sin2 φ cos2 θ+ρ2 sin2 φ sin2 θ+ρ2 cos2 φ) 3 2
ρ 2 sin φ dρ dθ dφ
� 2π � 1
e
0 0
(ρ2 sin2 φ+ρ2 cos2 φ) 3 2 2
ρ sin φ dρ dθ dφ
� π � 2π � 1
=
e
0 0 0
ρ3
ρ 2 sin φ dρ dθ dφ = · · ·
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, V
Example (continued)
� π
· · · =
0
� 2π � 1
0
0
e ρ3
ρ 2 sin φ dρ dθ dφ
� π
= 2π
0
� π
= 2π sin φ
0
�� 1
= 2π e
0
ρ3
ρ 2 dρ
� 1
e
0
ρ3
ρ 2 sin φ dρ dφ
�� 1
e
0
ρ3
ρ 2 �
dρ dφ
� 1
� �� π �
sin φ dφ = 4π
0
= 4π
3
� 1
0
0
e ρ3
ρ 2 dρ
e u du = 4π
(e − 1).
3
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, VI
Example
Let H be the upper hemisphere above the xy-plane and below
the sphere x 2 + y 2 + z 2 = 1, i.e.,
H = {(x, y, z) : x 2 + y 2 + z 2 ≤ 1, z ≥ 0}.
What is ���
H x 2 + y 2 dV ?
In spherical coordinates:
�
�
H = (ρ, θ, φ) : ρ ∈ [0, 1], θ ∈ [0, 2π], φ ∈ 0, π
��
.
2
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, VII
Example (continued)
Then:
���
H
=
=
x 2 + y 2 dV
� 2π
0
� 2π
0
� π
2
0
� π
2
0
� 1
0
� 1
0
=
(ρ 2 cos 2 θ sin 2 φ + ρ 2 sin 2 θ sin 2 φ)
ρ 2 sin φ dρ dφ dθ
ρ 4 (cos 2 θ + sin 2 θ) sin 3 φ dρ dφ dθ
� 2π
0
� π
2
0
� 1
0
ρ 4 sin 3 φ dρ dφ dθ = · · ·
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, VIII
Example (continued)
· · · =
� 2π
0
= 2π
5
� π
2
0
= 2π
� π
2
0
= − 2π
5
� 1
ρ 4 sin 3 φ dρ dφ dθ
0
�� 1
0
ρ 4 �
dρ
�� π
2
sin 3 φ dφ = 2π
5
� 0
1
0
� π
2
0
sin 3 φ dφ
�
sin φ(1 − cos 2 φ) dφ
� 1
1 − u 2 du = 2π
1 − u
5 0
2 du
= 2π
�
u −
5
u3
� �
u=1 �
�
3 �
u=0
= 4π
15 .
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, IX
Example
Determine the volume V of the solid E that lies above the
cone z = � x 2 + y 2 and below the sphere x 2 + y 2 + z2 = z.
Note:
x 2 + y 2 + z 2 = z ⇐⇒ x 2 + y 2 �
+ z − 1
�2 =
2
1
4 .
Project E onto the xy-plane and obtain D:
�
D = (x, y) : x 2 + y 2 ≤ 1
�
4
� �
= (r, θ) : r ∈ 0, 1
�
�
, θ ∈ [0, 2π] .
2
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, X
Example (continued)
Note:
(x, y, z) ∈ E ⇐⇒
(x, y) ∈ D and � x 2 + y 2 ≤ z ≤
Hence, in cylindrical coordinates:
E =
�
�
(r, θ, z) : r ∈ 0, 1
�
, θ ∈ [0, 2π],
2
r ≤ z ≤
� 1
4 − x 2 − y 2 + 1
2
�
1
4 − r 2 + 1
�
2
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XI
Example (continued)
Then:
���
V =
E
1 dV
=
= 2π
� 2π
� 1
2
0 0
� 1
2
0
= 2π
� � 1
4 −r 2 + 1
2
r dz dr dθ
r
��
1
4 − r 2 + 1
�
− r r dr
2
�� 1
� 1
2
2
0
r
� 1
4 − r 2 dr +
0
r
2 − r 2 �
dr .
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XII
Example (continued)
On the side:
and
� 1
2
0
r
� 1
2
0
r
2 − r 2 dr =
r 2
4
� 1
4 − r 2 dr = − 1
2
− r 3
3
� 0
1
4
= 1
2
�
�
�
�
r= 1
2
r=0
√ u du
� 1
4
0
= 1 1 1
− =
16 24 48
√ u du = u 3
2
3
�
�
�
�
�
u= 1
4
u=0
= 1
24 .
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XIII
Example (continued)
Therefore:
V = 2π
� � 1
2
�
1
r
0 4 − r 2 dr +
� �
1 1
= 2π +
24 48
= π
8 .
� 1
2
0
r
2 − r 2 �
dr
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XIV
Example (continued)
Pass to spherical coordinates.
For (x, y, z) on the sphere:
ρ cos φ = z = x 2 + y 2 + z 2
= ρ 2 cos 2 θ sin 2 φ + ρ 2 sin 2 θ sin 2 φ + ρ 2 cos 2 φ = ρ 2 .
Hence, the sphere is:
�
�
(ρ, θ, φ) : θ ∈ [0, 2π], φ ∈
0, π
2
� �
, ρ = cos φ .
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XV
Example (continued)
For (x, y, z) on the cone:
ρ cos φ = z = � x 2 + y 2
�
= ρ2 cos2 θ sin2 φ + ρ2 sin2 θ sin2 φ = ρ sin φ.
Hence, the cone is:
{(ρ, θ, φ) : ρ ≥ 0, θ ∈ [0, 2π], φ ∈ [0, π], cos φ = sin φ}
�
= (ρ, θ, φ) : ρ ≥ 0, θ ∈ [0, 2π], φ = π
�
.
4
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XV
Example (continued)
It follows that
�
�
E = (ρ, θ, φ) : θ ∈ [0, 2π], φ ∈ 0, π
�
�
, 0 ≤ ρ ≤ cos φ .
4
Hence:
���
V =
E
1 dV =
= 2π
�
= 2π
� π
4
0
sin φ
� 2π
0
� π
4
0
ρ 3
3
� π
4
� cos φ
0 0
�� cos φ
sin φ
0
� �
ρ=cos φ �
�
� dφ = 2π
3
ρ=0
ρ 2 sin φ dρ dφ dθ
ρ 2 �
dρ dφ
� 1
4
0
sin φ cos 3 φ dφ
= · · ·
MATH 209—
Calculus,
III
Volker Runde
Triple
integrals in
cylindrical
coordinates
Triple
integrals in
spherical
coordinates
Examples, XVI
Example (continued)
· · · = 2π
3
= 2π
3
� 1
4
u= 1
√
2
� 1
√2
sin φ cos
0
3 φ dφ = − 2π
u
3 1
3 du
⎛
⎝ u4
⎞
�u=1
�
� ⎠
4 � = 2π
� �
1 1
− =
3 4 16
2π
3
3 π
=
16 8 .
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