Glossary

9.1 Definition and properties of double integrals

3 min read•august 6, 2024

Double integrals extend the concept of integration to functions of two variables. They calculate the over a in the xy-plane, representing a powerful tool for analyzing three-dimensional spaces and their properties.

This section introduces the definition and key properties of double integrals. We'll learn how to set up and evaluate these integrals, understand their geometric interpretation, and explore important theorems that make working with them easier and more intuitive.

Definition and Properties

Double Integral and Rectangular Region

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  • represents the volume under a surface z=f(x,y)z=f(x,y) over a rectangular region RR in the xyxy-plane
  • Rectangular region R=[a,b]×[c,d]R=[a,b]\times[c,d] is the Cartesian product of two closed intervals [a,b][a,b] and [c,d][c,d]
    • Can be visualized as a rectangle with sides parallel to the coordinate axes
    • Defined by the inequalities a≤x≤ba\leq x\leq b and c≤y≤dc\leq y\leq d
  • Double integral of a function f(x,y)f(x,y) over a rectangular region RR is denoted as ∬Rf(x,y)dA\iint_R f(x,y) dA
    • dAdA represents the area element in the xyxy-plane
    • Geometrically, the double integral gives the volume of the solid bounded by the surface z=f(x,y)z=f(x,y) and the region RR

Integrable Function and Properties

  • Function f(x,y)f(x,y) is integrable over a rectangular region RR if the double integral ∬Rf(x,y)dA\iint_R f(x,y) dA exists
    • Integrability implies the function is bounded and has a finite number of discontinuities in RR
  • Additive property states that if f(x,y)f(x,y) is integrable over rectangular regions R1R_1 and R2R_2 that do not overlap except possibly along their boundaries, then f(x,y)f(x,y) is integrable over R1∪R2R_1\cup R_2 and ∬R1∪R2f(x,y)dA=∬R1f(x,y)dA+∬R2f(x,y)dA\iint_{R_1\cup R_2} f(x,y) dA = \iint_{R_1} f(x,y) dA + \iint_{R_2} f(x,y) dA
    • Allows the division of a region into smaller subregions for easier computation
  • Linearity property states that if f(x,y)f(x,y) and g(x,y)g(x,y) are integrable over a rectangular region RR and cc is a constant, then cf(x,y)cf(x,y) and f(x,y)+g(x,y)f(x,y)+g(x,y) are also integrable over RR and:
    • ∬Rcf(x,y)dA=c∬Rf(x,y)dA\iint_R cf(x,y) dA = c\iint_R f(x,y) dA
    • ∬R[f(x,y)+g(x,y)]dA=∬Rf(x,y)dA+∬Rg(x,y)dA\iint_R [f(x,y)+g(x,y)] dA = \iint_R f(x,y) dA + \iint_R g(x,y) dA

Riemann Sums

Approximating Double Integrals

  • Riemann sum approximates the double integral by partitioning the rectangular region RR into smaller rectangles and summing the volumes of rectangular prisms
  • Partition the intervals [a,b][a,b] and [c,d][c,d] into mm and nn subintervals of equal width Δx=b−am\Delta x = \frac{b-a}{m} and Δy=d−cn\Delta y = \frac{d-c}{n}, respectively
    • This creates mnmn subrectangles RijR_{ij} with dimensions Δx\Delta x by Δy\Delta y
  • Choose a sample point (xij∗,yij∗)(x_{ij}^*,y_{ij}^*) within each subrectangle RijR_{ij}
  • Riemann sum is given by ∑i=1m∑j=1nf(xij∗,yij∗)ΔxΔy\sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*) \Delta x \Delta y
    • Represents the sum of the volumes of rectangular prisms with base area ΔxΔy\Delta x \Delta y and height f(xij∗,yij∗)f(x_{ij}^*,y_{ij}^*)

Limit of Riemann Sum

  • As the number of partitions mm and nn approach infinity, the Riemann sum approaches the exact value of the double integral
  • Limit of the Riemann sum is written as lim⁡m,n→∞∑i=1m∑j=1nf(xij∗,yij∗)ΔxΔy=∬Rf(x,y)dA\lim_{m,n\to\infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*) \Delta x \Delta y = \iint_R f(x,y) dA
    • This limit exists if and only if the function f(x,y)f(x,y) is integrable over the rectangular region RR
  • The choice of sample points (xij∗,yij∗)(x_{ij}^*,y_{ij}^*) does not affect the limit as long as they lie within their respective subrectangles RijR_{ij}
    • Common choices include the midpoint, lower left corner, or upper right corner of each subrectangle

Theorems

Comparison Theorem for Double Integrals

  • Comparison theorem states that if f(x,y)≤g(x,y)f(x,y) \leq g(x,y) for all (x,y)(x,y) in a rectangular region RR, then ∬Rf(x,y)dA≤∬Rg(x,y)dA\iint_R f(x,y) dA \leq \iint_R g(x,y) dA
    • Provides a way to estimate or bound the value of a double integral
  • If f(x,y)≤g(x,y)≤h(x,y)f(x,y) \leq g(x,y) \leq h(x,y) for all (x,y)(x,y) in RR, then ∬Rf(x,y)dA≤∬Rg(x,y)dA≤∬Rh(x,y)dA\iint_R f(x,y) dA \leq \iint_R g(x,y) dA \leq \iint_R h(x,y) dA
    • Sandwich theorem for double integrals
  • Useful when the exact value of a double integral is difficult to compute but can be bounded by simpler functions
    • For example, if 0≤f(x,y)≤10 \leq f(x,y) \leq 1 over a rectangular region RR with area AA, then 0≤∬Rf(x,y)dA≤A0 \leq \iint_R f(x,y) dA \leq A

Key Terms to Review (18)

∬: The symbol ∬ represents a double integral, which is a mathematical operation used to calculate the volume under a surface in a two-dimensional region. This concept extends the idea of single integrals to higher dimensions, allowing for the evaluation of integrals over two variables simultaneously. Double integrals play a critical role in various applications, such as calculating areas, volumes, and averages over regions in the xy-plane.
Absolute convergence: Absolute convergence refers to the property of a series where the series of the absolute values of its terms converges. If a series is absolutely convergent, it implies that the original series converges as well, and this property is crucial for evaluating double integrals and understanding their behavior over rectangles, especially when interchanging the order of integration.
Boundedness of Functions: Boundedness of functions refers to a property where a function's values stay within a specific range, meaning there are upper and lower limits to the outputs over its entire domain. This concept is crucial for understanding the behavior of functions, especially when considering integration and convergence in multiple dimensions.
Change of Variables: Change of variables is a mathematical technique used to simplify complex integrals by transforming the variables of integration to a new set that makes evaluation easier. This technique is crucial when working with multiple integrals, allowing for the conversion between different coordinate systems and facilitating calculations in various contexts.
Conditional Convergence: Conditional convergence refers to the property of a series where it converges, but does so only when the terms are summed in a specific order. In the context of double integrals, understanding conditional convergence is crucial because it can affect the value of the integral when changing the order of integration. This concept highlights that not all convergent series behave the same way, especially when dealing with infinite sums or integrals.
Double integral: A double integral is a mathematical operation used to compute the volume under a surface in three-dimensional space, defined by a function of two variables over a specified region. This operation extends the concept of a single integral, allowing for the integration of functions across two dimensions, thereby enabling the calculation of areas, volumes, and other properties of two-variable functions.
Dx dy: The notation 'dx dy' represents the infinitesimal area element used in double integrals. It signifies the integration process over a two-dimensional region, where 'dx' indicates a small change in the x-direction and 'dy' represents a small change in the y-direction. This concept is crucial for calculating areas and volumes in multiple dimensions, enabling transformations and evaluations of integrals in different coordinate systems.
Fubini's Theorem: Fubini's Theorem states that if a function is continuous over a rectangular region, then the double integral of that function can be computed as an iterated integral. This theorem allows for the evaluation of double integrals by integrating one variable at a time, simplifying the process significantly. It's essential for understanding how to compute integrals over more complex regions and dimensions.
Iterated Integral: An iterated integral is a method used to compute multiple integrals by breaking them down into simpler, sequential integration processes. This approach allows for the evaluation of double or triple integrals by integrating one variable at a time while treating the others as constants. The concept is foundational in evaluating integrals over two or three-dimensional regions, linking together the definition and properties of integrals in higher dimensions.
Jacobian Determinant: The Jacobian determinant is a scalar value that represents the rate of change of a function with respect to its variables, particularly when transforming coordinates from one system to another. It is crucial for understanding how volume and area scale under these transformations, and it plays a significant role in evaluating integrals across different coordinate systems.
Lebesgue's Dominated Convergence Theorem: Lebesgue's Dominated Convergence Theorem is a fundamental result in measure theory that allows for the interchange of limit and integration under certain conditions. It states that if a sequence of measurable functions converges almost everywhere to a limit function, and if there exists an integrable function that dominates the sequence, then the limit of the integrals equals the integral of the limit function. This theorem is crucial in understanding how limits and integrals interact, especially in the context of double integrals.
Lower Limit: In the context of double integrals, the lower limit refers to the minimum value of the variable of integration within the specified bounds of the integral. It plays a critical role in defining the area or volume being calculated by indicating where the integration starts along the respective axis. Understanding the lower limit is essential for accurately evaluating double integrals, as it directly influences the region of integration and the final result.
Mass distribution: Mass distribution refers to how mass is spread out in a given region or volume, often described mathematically using density functions. Understanding mass distribution is crucial for calculating properties such as total mass, center of mass, and the effects of gravitational forces in various physical scenarios.
Polar coordinates: Polar coordinates are a two-dimensional coordinate system that uses the distance from a reference point (the origin) and an angle from a reference direction to uniquely determine the position of a point in the plane. This system is particularly useful for problems involving circular or rotational symmetry, allowing for simpler integration and analysis in certain contexts.
Rectangular region: A rectangular region is a specific area in a two-dimensional space defined by the Cartesian product of two intervals. This region serves as the foundational concept in calculating double integrals and can be extended to three-dimensional space for triple integrals, where it becomes a rectangular prism. The properties of a rectangular region help in establishing limits for integration, making it easier to compute areas or volumes under curves and surfaces.
Tonelli's Theorem: Tonelli's Theorem is a fundamental result in measure theory that provides conditions under which the integral of a non-negative function over a product space can be computed as an iterated integral. This theorem is particularly important for evaluating double integrals, as it allows for interchanging the order of integration when dealing with non-negative functions. Understanding this theorem is crucial for grasping the properties of double integrals and the process of iterated integration.
Upper Limit: In the context of double integrals, the upper limit refers to the boundary or constraint that defines the maximum value for a variable in the integration process. This concept is crucial when determining the region of integration and affects how functions are evaluated over a given area. The upper limit helps in establishing the scope of integration for double integrals, allowing for accurate calculations of areas and volumes.
Volume under a surface: Volume under a surface refers to the three-dimensional space occupied beneath a function defined over a specific region in the xy-plane. This concept is crucial in understanding how to calculate the total volume formed by a surface, particularly when using double integrals to sum infinitesimal volume elements across the region of interest, providing insight into the geometric interpretation of integrals.
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