Double Integrals

4 Key excerpts on "Double Integrals"

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  eBook - PDF

  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  TECHNOLOGY MASTERY If you have a CAS with a built-in capa- bility for computing iterated Double Integrals, use it to check Example 3. Example 4 Evaluate the double integral ∫ ∫ R y 2 x dA over the rectangle R = {(x, y) : −3 ≤ x ≤ 2, 0 ≤ y ≤ 1}. Theorem 14.1.3 guarantees that the double integral in Example 4 can also be evaluated by integrating first with respect to y and then with respect to x. Verify this. Solution. In view of Theorem 14.1.3, the value of the double integral can be obtained by evaluating one of two possible iterated Double Integrals. We choose to integrate first with respect to x and then with respect to y. ∫ ∫ R y 2 x dA = ∫ 1 0 ∫ 2 −3 y 2 x dx dy = ∫ 1 0 [ 1 2 y 2 x 2 ] 2 x=−3 dy = ∫ 1 0 ( − 5 2 y 2 ) dy = − 5 6 y 3 ] 1 0 = − 5 6 The integral in Example 4 can be interpreted as the net signed volume between the rectangle [−3, 2] × [0, 1] and the surface z = y 2 x. That is, it is the volume below z = y 2 x and above [0, 2] × [0, 1] minus the volume above z = y 2 x and below [−3, 0] × [0, 1] (Figure 14.1.7). 2 −3 (−3, 1) (2, 1) y z x z = y 2 x on [−3, 2] × [0, 1] Figure 14.1.7 PROPERTIES OF Double Integrals To distinguish between Double Integrals of functions of two variables and definite integrals of functions of one variable, we will refer to the latter as single integrals. Because Double Integrals, like single integrals, are defined as limits, they inherit many of the properties of limits. The following results, which we state without proof, are analogs of those in Theo- rem 5.5.4. ∫ ∫ R cf (x, y) dA = c ∫ ∫ R f (x, y) dA (c a constant) (9) ∫ ∫ R [ f (x, y) + g(x, y)] dA = ∫ ∫ R f (x, y) dA + ∫ ∫ R g(x, y) dA (10) ∫ ∫ R [ f (x, y) − g(x, y)] dA = ∫ ∫ R f (x, y) dA − ∫ ∫ R g(x, y) dA (11) y x z z = f ( x, y) R 1 R 2 R The volume of the entire solid is the sum of the volumes of the solids above R 1 and R 2 .

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  eBook - PDF

  Guide to Mathematical Methods

  • John Gilbert, Camilla Jordan, David A Towers(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  CHAPTER 12 Double Integrals By the end of this chapter you will have • been introduced to the idea of a double integral; • learnt how to use a Jacobian in changing variables; • read about Green’s Theorem. Aims and Objectives 12.1 Double Integrals Recall that in Section 6.1 we defined the definite integral of a function f as the area under the graph of f . This was computed as the limit of a sum of areas of rectangles, whose width became vanishingly small. We now carry out a similar construction for the double integral of a function f of two variables. In this case, we start with a region R of area A of the xy plane, and construct a cylinder with R as its base, and with its axis parallel to the z axis, as shown in Figure 12.1. Then the integral of f over the region R is the volume of the cylinder between the xy plane and the surface z = f ( x, y ). In order to compute this volume, we divide R into a number of subregions, the i ’th of which, δR i , has area δA i , and construct cylinders on each of these as a base, with axes parallel to the z axis. If ( x i , y i ) is a point inside the subregion δR i , then the volume between this and the surface z = f ( x, y ) is approximately f ( x i , y i ) δA i . Summing over all these subregions and proceeding to the limit as δA i → 0, which requires the number of subregions to increase infinitely to maintain the equality i δA i = A, 305 306 Guide to Mathematical Methods δR i z = f ( x, y ) Figure 12.1: A cylinder under the surface z = f ( x, y ) we obtain the double integral of the function f over the region R as R f d A = lim max δA i → 0 i f ( x i , y i ) δA i . We assume that the errors of approximating the cylinder over R by cylinders with flat tops disappear in the limit. We say that f is integrable if the limit exists. It can be shown to exist for suitably smooth functions. We now turn to a method of evaluating such integrals.

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  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  10 Double Integrals 10.1 Double Integrals Over Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 10.2 Double Integrals Over Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 10.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 10.4 Applications of Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Chapter 10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Chapter 10 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 10.1 Double Integrals Over Rectangles In Chapter 5, we saw that the definition of the definite integral was motivated by finding areas of regions under curves. In much the same way, our desire to find volumes of solids under surfaces in this section will lead us to the definition of what is known as a double integral. Volumes and Double Integrals Suppose f is a continuous function in x and y that is defined over the rectangle R given by Figure 10.1 Figure 10.2 R = [a, b] × [c, d] = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d} If f (x, y) ≥ 0 for all (x, y) in R, then the graph of z = f (x, y) is a surface that lies on or above the xy-plane. Our goal is to find the volume V of the solid that lies below the graph of z = f (x, y) and above the region R. See Figure 10.1 for an illustration of such a solid. We begin by approximating V using volumes of thin rectangular boxes. To do so, we first partition R into a grid of n small rectangles using lines parallel to the x- and y-axes, as shown in Figure 10.2. After ordering the rectangles in a convenient fashion (perhaps by “columns” in the y-direction within “rows” in the x-direction), we denote the ith rectangle by R i and denote its area by ΔA i . Within each R i we choose an arbitrary sample point (x i , y i ) (Figure 10.3).

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  eBook - ePub

  Advanced Calculus

  Differential Calculus and Stokes' Theorem

  • Pietro-Luciano Buono(Author)
  • 2016(Publication Date)
  • De Gruyter

    (Publisher)

  As mentioned above, we also need a more effective way of computing the double integral than using the definition. We know from elementary calculus that for simple integrals of functions of one variable, the Fundamental Theorem of Calculus enables one to relate the value of the integral to the existence of an antiderivative, or indefinite integral. There is no such result for Double Integrals. However, it is possible to decompose the double integral into a succession of two simple integrals as the next result shows.

  Theorem 7.2.4 (Fubini). If f is integrable on the rectangle ℝ. Then

  The idea of this theorem is that one can decompose the rectangle [a 0 , a 1 ] × [b 0 , b 1 ] using slices parallel to one of the axes. This is done as follows, say for slices parallel to the y -axis. Fix a value x ∈ [a 0 , a 1 ] and consider f (x, y ) with y ∈ [b 0 , b 1 ]. Then, we can compute the integral of the slice obtained using f (x, y ):

  Now, (7.2) depends on the value of x chosen to obtain the slice and so the result

  Fig. 7.6. Slice at x = x 0 of the domain.

  of (7.2) is a function of x . We write

  Thus, for every x ∈ [a 0 , a 1 ], G (x ) is the integral of a slice. Fubini’s theorem tells us that summing up all the integrals G (x ) for x ∈ [a 0 , a 1 ] corresponds to the integral of f (x, y ) over R .

  Example 7.2.5. Let ℝ = [0, 3] × [−1, 1] and f (x, y ) =

  ex

  +

  y . We compute

  Computation of mass and populations

  The double integral can be used to compute physical quantities such as total mass or total electric charge if the mass or charge density is known. Suppose that a rectangular plate R of a certain material has density (with units of mass/unit area, say kg/m 2 ) given by a continuous function ρ (x, y

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