Triple integrals in cylindrical and spherical coordinates

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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 �


�� 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


3

3 π

=

16 8 .

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