Chapter 17: Double and Triple Integrals

Section 17.1 Multiple Sigma Notation

a. Double Sigma Notation

b. Properties

Section 17.2 Double Integrals

a. Double Integral Over a Rectangle

b. Partitions

c. More on Partitions

d. Upper Sums and Lower Sums

e. Double Integral Over a Rectangle R

f. Double Integral as a Volume

g. Volume of T

h. Double Integral Over a Region

i. Volume of the Solid T

j. Elementary Properties: I and II

k. Elementary Property III

l. Elementary Property IV

m. Mean-Value Theorem for Double Integrals

Section 17.3 The Evaluation of Double Integrals By Repeated Integrals

a. Type I Regions

b. Type II Region

c. Reduction Formulas Viewed Geometrically

d. Reduction Formula

e. Symmetry in Double Integration

Section 17.4 The Double Integral as a Limit of Riemann Sums; Polar Coordinates

a. Limit of Riemann Sums

Chapter 17: Double and Triple Integrals

Section 17.5 Further Applications of the Double Integration

a. Mass of a Plate

b. Center of Mass of a Plate

c. Centroids

d. Applications

Section 17.6 Triple Integrals

a. Triple Integral Over a Box

b. Triple Integral Over a More General Solid

c. Volume

Section 17.7 Reduction to Repeated Integrals

a. Formula and Illustration

Section 17.8 Cylindrical Coordinates

a. Rectangular Coordinates/ Cylindrical Coordinates

b. Evaluating Triple Integrals

c. Volume in Cylindrical Coordinates Section 17.9 Spherical Coordinates

a. Longitude, Colatitude, Latitude

b. Spherical Wedge

c. Volume

Section 17.10 Jacobians, Changing Variables

a. Change of Variables

b. Jacobian

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Multiple Sigma Notation

When two indices are involved, say,

we use double-sigma notation. By

we mean the sum of all the a_ij where i ranges from 1 to m and j ranges from 1 to n. For example,

( )

2 5 , 2 , 1

5

i j j

ij ij ij

a a i a i

= = j = +

+

3 2

2 2 2 2 3 3 2

1 1

2 5

^{i j}

2 5 2 5 2 5 2 5 2 5 2 5 420

i= j=

= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ =

∑∑

Multiple Sigma Notation

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Double Integrals

The Double Integral over a Rectangle

We start with a function f continuous on a rectangle R : a ≤ x ≤ b, c ≤ y ≤ d We want to define the double integral of f over R:

( )

^{, .}

R

f x y dx dy

∫∫

Double Integrals

First we explain what we mean by a partition of the rectangle R. We begin with a partition

P₁ = {x₀, x₁, . . . , x_m} of [a, b] , and a partition

P₂ = {y₀, y₁, . . . , y_n} of [c, d] . The set

P = P₁ × P₂ = {(x_i , y_j) : x_i ∈ P₁, y_j ∈ P₂}

is called a partition of R. The set P consists of all the grid points (x_i , y_j ).

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Double Integrals

Using the partition P, we break up R into m × n nonoverlapping rectangles R_ij : x_i−1 ≤ x ≤ x_i , y_j−1 ≤ y ≤ y_j , where 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Double Integrals

The sum of all the products

is called the P upper sum for f :

The sum of all the products

is called the P lower sum for f :

(

area of

) ⁽

1

⁾ (

1

)

i j i j i j i i j j i j i j

M R = M x − x₋ y − y ₋ = M ∆ ∆x y

(

^{area of}

) ⁽

¹

⁾ (

¹

)

i j i j i j i i j j i j i j

m R = m x −x₋ y − y ₋ = m ∆ ∆x y

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Double Integrals

Double Integrals

The Double Integral as a Volume

If f is continuous and nonnegative on the rectangle R, the equation z = f (x, y)

represents a surface that lies above R. In this case the double integral

gives the volume of the solid that is bounded below by R and bounded above by the surface z = f (x, y).

( )

R

f x dx dy

∫∫

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Double Integrals

Since the choice of a partition P is arbitrary, the volume of T must be the double integral:

The double integral

gives the volume of a solid of constant height 1 erected over R. In square units this is just the area of R:

1

R R

dx dy = dx dy

∫∫ ∫∫

Double Integrals

The Double Integral over a Region

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Double Integrals

If f is continuous and nonnegative over Ω, the extended f is nonnegative on all of R. The volume of the solid T bounded above by z = f (x, y) and

bounded below by Ω is given by:

Double Integrals

Four Elementary Properties of the Double Integral:

The Ω referred to is a basic region. The functions f and g are assumed to be continuous on Ω.

I. Linearity: The double integral of a linear combination is the linear combination of the double integrals:

II. Order: The double integral preserves order:

if f ≥ 0 on Ω, then

if f ≤ g on Ω, then

( )

^,

( )

^,

( )

^,

( )

^,

f x y g x y dx dy f x y dx dy g x y dx dy

α β α β

Ω Ω Ω

 +  = +

 

∫∫ ∫∫ ∫∫

( )

^{, 0}

f x y dx dy

Ω

∫∫

≥

( )

^,

( )

^,

f x y dx dy g x y dx dy

Ω Ω

∫∫

≤

∫∫

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Double Integrals

III. Additivity: If Ω is broken up into a finite number of nonoverlapping basic regions Ω₁, . . . , Ω_n, then

( ) ( ) ( )

1

, , ,

n

f x y dx dy f x y dx dy f x y dx dy

Ω Ω Ω

= + +

∫∫ ∫∫

^

∫∫

Double Integrals

( ) , (

0

,

0

) ( area of )

f x y dx dy f x y

Ω

= ⋅ Ω

∫∫

IV. Mean-value condition: There is a point (x₀, y₀) in Ω for which

We call f (x₀, y₀) the average value of f on Ω .

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Double Integrals

The Evaluation of Double Integrals By Repeated Integrals

Type I Region The projection of Ω onto the x-axis is a closed interval [a, b] and Ω consists of all points (x, y) with

a ≤ x ≤ b and φ1

( )

x ≤ ≤y φ2

( )

x .

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The Evaluation of Double Integrals By Repeated Integrals

Type II Region The projection of Ω onto the y-axis is a closed interval [c, d] and Ω consists of all points (x, y) with

c ≤ y ≤ d and ψ₁(y) ≤ x ≤ ψ₂(y).

In this case

The Evaluation of Double Integrals By Repeated Integrals

The Reduction Formulas Viewed Geometrically

Suppose that f is nonnegative and Ω is a region of Type I. The double integral over Ω gives the volume of the solid T bounded above by the surface z = f (x, y) and bounded below by the region Ω:

( )

¹ f x y dx dy

( )

^, volume of T

Ω

∫∫

=

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The Evaluation of Double Integrals By Repeated Integrals

( ) ( )

( )

2( )

1

, ,

b x

a x

f x y dx dy f x y dy dx

φ φ Ω

 

 

=  

 

∫∫ ∫ ∫

We can also calculate the volume of T by the method of parallel cross sections.

Combining (1) with (2), we have the first reduction formula

( ) ( )

( )

2( )

1

2 , volume of

b x

a x

f x y dy dx T

φ φ

 

  =

 

 

∫ ∫

The other reduction formula can be obtained in a similar manner.

The Evaluation of Double Integrals By Repeated Integrals

Suppose that Ω is symmetric about the y-axis.

If f is odd in x [ f (−x, y) = −f (x, y)], then If f is even in x [ f (−x, y) = f (x, y)], then

Suppose that Ω is symmetric about the x-axis.

If f is odd in y [ f (x,−y) = −f (x, y)], then If f is even in y [ f (x,−y) = f (x, y)], then Symmetry in Double Integration

( )

^{, 0}

f x y dx dy

Ω

∫∫

=

( ) ( )

right half of

, 2 ,

f x y dx dy f x y dx dy

Ω Ω

∫∫

=

∫∫

( )

^{, 0}

f x y dx dy

Ω

∫∫

=

( ) ( )

upper half of

, 2 ,

f x y dx dy f x y dx dy

Ω Ω

∫∫

=

∫∫

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The Double Integral as a Limit of Riemann Sums; Polar Coordinates

The Double Integral as a Limit of Riemann Sums; Polar Coordinates

Evaluating Double Integrals Using Polar Coordinates

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The Double Integral as a Limit of Riemann Sums; Polar Coordinates

The Double Integral as a Limit of Riemann Sums; Polar Coordinates

The function f (x) = e^−x2 has no elementary antiderivative. Nevertheless, by taking a circuitous route and then using polar coordinates, we can show that

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Further Applications of the Double Integral

A thin plane distribution of matter (we call it a plate) is laid out in the xy-plane in the form of a basic region Ω. If the mass density of the plate (the mass per unit area) is a constant λ, then the total mass M of the plate is simply the density λ times the area of the plate:

M = λ × the area of Ω.

If the density varies continuously from point to point, say λ = λ(x, y), then the mass of the plate is the average density of the plate times the area of the plate:

M = average density × the area of Ω.

This is a double integral:

Further Applications of the Double Integral

The Center of Mass of a Plate

The center of mass x_M of a rod is a density-weighted average of position taken over the interval occupied by the rod:

The coordinates of the center of mass of a plate (x_M, y_M) are determined by two density weighted averages of position, each taken over the region occupied by the plate:

( )

^.

b

M a

x M =

∫

xλ x dx

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Further Applications of the Double Integral

Centroids

If the plate is homogeneous, then the mass density λ is constantly M/A where A is the area of the base region Ω. In this case the center of mass of the plate falls on the centroid of the base region (a notion with which you are already familiar). The centroid depends only on the geometry of

(

^{x y,}

)

Ω:

Further Applications of the Double Integral

Kinetic Energy and Moment of Inertia

The Moment of Inertia of a Plate

Radius of Gyration

The Parallel Axis Theorem

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Triple Integrals

Triple Integrals

The Triple Integral Over a More General Solid

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Triple Integrals

Volume as a Triple Integral

The simplest triple integral of interest is the triple integral of the function that is constantly 1 on T . This gives the volume of T :

Reduction to Repeated Integrals

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Cylindrical Coordinates

The cylindrical coordinates (r, θ, z) of a point P in xyz-space are shown geometrically in Figure 17.8.1. The first two coordinates, r and θ, are the usual plane polar coordinates except that r is taken to be nonnegative and θ is restricted to the interval [0, 2π]. The third coordinate is the third rectangular coordinate z.

In rectangular coordinates, the coordinate surfaces

x = x₀, y = y₀, z = z₀

are three mutually perpendicular planes. In cylindrical coordinates, the coordinate surfaces take the form

r = r₀, θ = θ₀, z = z₀

Cylindrical Coordinates

Evaluating Triple Integrals Using Cylindrical Coordinates

Suppose that T is some basic solid in xyz-space, not necessarily a wedge. If T is the set of all (x, y, z) with cylindrical coordinates in some basic solid S in rθz-space, then

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Cylindrical Coordinates

Volume Formula

If f (x, y, z) = 1 for all (x, y, z) in T , then (17.8.1) reduces to

The triple integral on the left is the volume of T . In summary, if T is a basic solid in xyz-space and the cylindrical coordinates of T constitute a basic solid S in rθz-space, then the volume of T is given by the formula

.

T S

dx dy dz = r dr d dz θ

∫∫∫ ∫∫∫

Spherical Coordinates

φ

The spherical coordinates (ρ, θ, ) of a point P in xyz-space are shown geometrically in Figure 17.9.1. The first coordinate ρ is the distance from P to the origin; thus ρ ≥ 0.

The second coordinate, the angle marked θ, is the second coordinate of cylindrical coordinates; θ ranges from 0 to 2π. We call θ the longitude. The third coordinate, the angle marked , ranges only from 0 to π. We call the colatitude, or more simply the polar angle. (The complement of would be the latitude on a globe.)

φ

φ φ

φ

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Spherical Coordinates

Spherical Coordinates

Volume Formula

Evaluating Triple Integrals Using Spherical Coordinates

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Jacobians; Changing Variables in Multiple Integration

Figure 17.10.1 shows a basic region Γ in a plane that we are calling the uv-plane. (In this plane we denote the abscissa of a point by u and the ordinate by v.) Suppose that

x = x(u, v), y = y(u, v) are continuously differentiable functions on the region Γ.

Jacobians; Changing Variables in Multiple Integration

is one-to-one on the interior of Γ, and the Jacobian

is never zero on the interior of Γ, then

As (u, v) ranges over Γ, the point (x, y), (x(u, v), y(u, v)) generates a region Ω in the xy-plane. If the mapping

(u, v) → (x, y)

( )

^,

x y

x y x y

u u

J u v

x y u v v u

v v

∂ ∂

∂ ∂ ∂ ∂

∂ ∂

= = −

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂