Chapter 17: Double and Triple Integrals

Transcription

1 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 b. Evaluating Double Integrals Using Polar Coordinates c. Double Integral Formulas d. Integrating e x 2 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 Jacobians, Changing Variables a. Change of Variables b. Jacobian

2 Multiple Sigma Notation When two indices are involved, say, we use double-sigma notation. By i j 2i aij = 25, aij =, aij = 1+ i 5 + j ( ) j 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, 3 2 i= 1 j= 1 = = i j

3 Multiple Sigma Notation

4 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.

5 Double Integrals First we explain what we mean by a partition of the rectangle R. We begin with a partition P 1 = {x 0, x 1,..., x m } of [a, b], and a partition P 2 = {y 0, y 1,..., y n } of [c, d]. The set P = P 1 P 2 = {(x i, y j ) : x i P 1, y j P 2 } is called a partition of R. The set P consists of all the grid points (x i, y j ).

6 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.

7 Double Integrals The sum of all the products is called the P upper sum for f : ( area of ) ( 1)( 1) M R = M x x y y = M x y ij ij ij i i j j ij i j The sum of all the products is called the P lower sum for f : ( area of ) ( 1)( 1) m R = m x x y y = m x y ij ij ij i i j j ij i j

8 Double Integrals

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

10 Double Integrals Since the choice of a partition P is arbitrary, the volume of T must be the double integral: The double integral R 1dx dy = R dx dy gives the volume of a solid of constant height 1 erected over R. In square units this is just the area of R:

11 Double Integrals The Double Integral over a Region

12 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:

13 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: (, ) (, ) (, ) (, ) α f x y + βg x y dx dy = α f x y dx dy + β g x y dx dy Ω Ω Ω II. Order: The double integral preserves order: if f 0 on Ω, then Ω f ( ) x, y dx dy 0 if f g on Ω, then Ω (, ) (, ) f x y dx dy g x y dx dy Ω

14 Double Integrals III. Additivity: If Ω is broken up into a finite number of nonoverlapping basic regions Ω 1,..., Ω n, then (, ) = (, ) + + (, ) f x y dx dy f x y dx dy f x y dx dy Ω Ω Ω 1 n

15 Double Integrals IV. Mean-value condition: There is a point (x 0, y 0 ) in Ω for which Ω (, ) = (, ) ( area of Ω) f x y dx dy f x y 0 0 We call f (x 0, y 0 ) the average value of f on Ω.

16 Double Integrals

17 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 φ ( x) y φ ( x) 1 2.

18 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 ψ 1 (y) x ψ 2 (y). In this case

19 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

20 The Evaluation of Double Integrals By Repeated Integrals We can also calculate the volume of T by the method of parallel cross sections. b φ 2 f x, y dy dx = volume of T a φ 1( x) 2 ( x) ( ) ( ) Combining (1) with (2), we have the first reduction formula Ω b φ 2( x ) f ( x, y) dx dy = f ( x, y) dy dx a φ 1( x) The other reduction formula can be obtained in a similar manner.

21 The Evaluation of Double Integrals By Repeated Integrals Symmetry in Double Integration 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 Ω Ω f ( ) x, y dx dy = 0 (, ) = 2 (, ) f x y dx dy f x y dx dy right half of Ω 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 Ω Ω f ( ) x, y dx dy = 0 (, ) = 2 (, ) f x y dx dy f x y dx dy upper half of Ω

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

23 The Double Integral as a Limit of Riemann Sums; Polar Coordinates Evaluating Double Integrals Using Polar Coordinates

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

25 The Double Integral as a Limit of Riemann Sums; Polar Coordinates The function f (x) = e x 2 has no elementary antiderivative. Nevertheless, by taking a circuitous route and then using polar coordinates, we can show that

26 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: This is a double integral: M = average density the area of Ω.

27 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: b ( ). xm M = xλ x dx a 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:

28 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 x, y depends only on the geometry of Ω: ( )

29 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

30 Triple Integrals

31 Triple Integrals The Triple Integral Over a More General Solid

32 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 :

33 Reduction to Repeated Integrals

34 Cylindrical Coordinates The cylindrical coordinates (r, θ, z) of a point P in xyz-space are shown geometrically in Figure 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 = x 0, y = y 0, z = z 0 are three mutually perpendicular planes. In cylindrical coordinates, the coordinate surfaces take the form r = r 0, θ = θ 0, z = z 0

35 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

36 Cylindrical Coordinates Volume Formula If f (x, y, z) = 1 for all (x, y, z) in T, then (17.8.1) reduces to T dx dy dz = r dr dθ dz. 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 S

37 Spherical Coordinates φ The spherical coordinates (ρ, θ, ) of a point P in xyz-space are shown geometrically in Figure 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 φ, ranges only from 0 to π. We call φ the colatitude, or more simply the polar angle. (The complement of φ would be the latitude on a globe.) φ

38 Spherical Coordinates

39 Spherical Coordinates Evaluating Triple Integrals Using Spherical Coordinates Volume Formula

40 Jacobians; Changing Variables in Multiple Integration Figure 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.

41 Jacobians; Changing Variables in Multiple Integration 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) is one-to-one on the interior of Γ, and the Jacobian (, ) J uv is never zero on the interior of Γ, then x y u u x y x y = = x y u v v u v v