Chapter 17: Double and Triple Integrals - math.ncku.edu.trchen/2018-2019 Teaching/Double and Triple...Section 17.4 The Double Integral as a Limit of Riemann Sums; Polar Coordinates

Main MenuSalas, Hille, Etgen Calculus: One and Several Variables

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Section 17.1 Multiple Sigma Notationa. Double Sigma Notationb. Properties

Section 17.2 Double Integralsa. Double Integral Over a Rectangleb. Partitionsc. More on Partitionsd. Upper Sums and Lower Sumse. Double Integral Over a Rectangle Rf. Double Integral as a Volumeg. Volume of Th. Double Integral Over a Regioni. Volume of the Solid Tj. Elementary Properties: I and IIk. Elementary Property IIIl. Elementary Property IVm. Mean-Value Theorem for Double Integrals

Section 17.3 The Evaluation of Double Integrals By Repeated Integralsa. Type I Regionsb. Type II Regionc. Reduction Formulas Viewed Geometricallyd. Reduction Formulae. Symmetry in Double Integration

Section 17.4 The Double Integral as a Limit of Riemann Sums; Polar Coordinatesa. Limit of Riemann Sumsb. Evaluating Double Integrals Using Polar Coordinatesc. Double Integral Formulasd. Integrating e−x2

Chapter 17: Double and Triple Integrals

Section 17.5 Further Applications of the Double Integrationa. Mass of a Plateb. Center of Mass of a Platec. Centroidsd. Applications

Section 17.6 Triple Integralsa. Triple Integral Over a Boxb. Triple Integral Over a More General Solidc. Volume

Section 17.7 Reduction to Repeated Integralsa. Formula and Illustration

Section 17.8 Cylindrical Coordinatesa. Rectangular Coordinates/ Cylindrical Coordinatesb. Evaluating Triple Integralsc. Volume in Cylindrical Coordinates

Section 17.9 Spherical Coordinatesa. Longitude, Colatitude, Latitudeb. Spherical Wedgec. Volume

Section 17.10 Jacobians, Changing Variablesa. Change of Variablesb. Jacobian



Multiple Sigma Notation

When two indices are involved, say,

we use double-sigma notation. By

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

( )22 5 , , 15

ji jij ij ij

ia a a ij

= = = ++

3 22 2 2 2 3 3 2

1 12 5 2 5 2 5 2 5 2 5 2 5 2 5 420i j

i j= =

= ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ =∑∑



Multiple Sigma Notation



Double Integrals

The Double Integral over a Rectangle

We start with a function f continuous on a rectangleR : 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 apartition

P1 = x0, x1, . . . , xm of [a, b] ,and a partition

P2 = y0, y1, . . . , yn of [c, d] .The set

P = P1 × P2 = (xi , yj) : xi ∈ P1, yj ∈ P2is called a partition of R. The set P consists of all the grid points (xi , yj ).



Double Integrals

Using the partition P, we break up R into m × n nonoverlapping rectanglesRij : xi−1 ≤ x ≤ xi , yj−1 ≤ y ≤ yj , where 1 ≤ i ≤ m, 1 ≤ j ≤ n.



Double IntegralsThe 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 :

( ) ( )( )1 1area of i j i j i j i i j j i j i jM R M x x y y M x y− −= − − = ∆ ∆

( ) ( )( )1 1area of i j i j i j i i j j i j i jm R m x x y y m x y− −= − − = ∆ ∆



Double Integrals



Double IntegralsThe Double Integral as a Volume

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

( )R

f x dx dy∫∫



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 isjust the area of R:

1R R

dx dy dx dy=∫∫ ∫∫



Double Integrals

The Double Integral over a Region



Double IntegralsIf 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 dyα β α βΩ Ω Ω

+ = + ∫∫ ∫∫ ∫∫

( ), 0f x y dx dyΩ

≥∫∫

( ) ( ), ,f x y dx dy g x y dx dyΩ Ω

≤∫∫ ∫∫



Double Integrals

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

( ) ( ) ( )1

, , ,n

f x y dx dy f x y dx dy f x y dx dyΩ Ω Ω

= + +∫∫ ∫∫ ∫∫



Double Integrals

( ) ( ) ( )0 0, , area of f x y dx dy f x yΩ

= ⋅ 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 .



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 ( ) ( )1 2 .x y xφ φ≤ ≤




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

c ≤ y ≤ d and ψ1(y) ≤ x ≤ ψ2(y). In this case




The Reduction Formulas Viewed GeometricallySuppose 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 , volume of f x y dx dy TΩ

=∫∫




( ) ( )( )

( )2

1

, ,xb

a x

f x y dx dy f x y dy dxφ

φΩ

=

∫∫ ∫ ∫

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 xb

a x

f x y dy dx Tφ

φ

= ∫ ∫

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




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

( ), 0f x y dx dyΩ

=∫∫( ) ( )

right halfof

, 2 ,f x y dx dy f x y dx dyΩ

Ω

=∫∫ ∫∫

( ), 0f x y dx dyΩ

=∫∫( ) ( )

upper halfof

, 2 ,f x y dx dy f x y dx dyΩ

Ω

=∫∫ ∫∫



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




Evaluating Double Integrals Using 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



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:




The Center of Mass of a PlateThe center of mass xM 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 (xM, yM) are determined by two density weighted averages of position, each taken over the region occupied by the plate:

( ) .b

M ax M x x dxλ= ∫




CentroidsIf the plate is homogeneous, then the mass density λ is constantly M/A where A isthe 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 Ω:( ),x y



Further Applications of the Double IntegralKinetic Energy and Moment of Inertia

The Moment of Inertia of a Plate

Radius of Gyration

The Parallel Axis Theorem



Triple Integrals



Triple IntegralsThe Triple Integral Over a More General Solid



Triple Integrals

Volume as a Triple IntegralThe 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



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




Evaluating Triple Integrals Using Cylindrical Coordinates

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




Volume FormulaIf 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 dzθ=∫∫∫ ∫∫∫



Spherical Coordinates

φ

The spherical coordinates (ρ, θ, ) of a point P in xyz-space are shown geometricallyin 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 cylindricalcoordinates; θ ranges from 0 to 2π. We call θ the longitude. The third coordinate, theangle 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

φφφ







Volume Formula

Evaluating Triple Integrals Using Spherical Coordinates



Jacobians; Changing Variables in Multiple IntegrationFigure 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 yx y x yu uJ u v

x y u v v uv v

∂ ∂∂ ∂ ∂ ∂∂ ∂= = −

∂ ∂ ∂ ∂ ∂ ∂∂ ∂

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Chapter 17: Double and Triple Integrals - math.ncku.edu.trchen/2018-2019 Teaching/Double and Triple...Section 17.4 The Double Integral as a Limit of Riemann Sums; Polar Coordinates