Glossary

4.1 Double Integrals over Rectangles

2 min read•july 25, 2024

extend single-variable integration to two dimensions, allowing us to calculate volumes under surfaces over rectangular regions. We use , applying to switch integration order when helpful.

These integrals are powerful tools for finding volumes and average values over rectangular regions. By choosing the right order of integration, we can simplify complex calculations and solve problems more efficiently.

Understanding Double Integrals over Rectangles

Evaluation of double integrals

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  • Double integrals extend single variable integration to two variables representing over rectangular region
  • Notation ∫∫Rf(x,y)dA\int\int_R f(x,y) dA where R denotes rectangular region
  • outer corresponds to one variable inner to other
  • Iterated integration process integrates wrt one variable treating other as constant then integrates result wrt second variable
  • Fubini's Theorem allows changing integration order facilitating simpler calculations (x-y to y-x)

Volume calculation with double integrals

  • Double integral geometrically represents volume under surface z=f(x,y)z = f(x,y) over region R
  • Volume calculation setup V=∫∫Rf(x,y)dAV = \int\int_R f(x,y) dA
  • Identify function f(x,y)f(x,y) representing surface (paraboloid, plane)
  • Determine rectangular region R boundaries (x from 0 to 2, y from 1 to 3)
  • Evaluate integral to find volume using iterated integration

Average value over rectangular regions

  • in two dimensions extends 1D concept to surfaces
  • Formula favg=1A∫∫Rf(x,y)dAf_{avg} = \frac{1}{A} \int\int_R f(x,y) dA
  • Calculate rectangular region area A=(b−a)(d−c)A = (b-a)(d-c) for rectangle [a,b]×[c,d][a,b] \times [c,d]
  • Set up and evaluate double integral of function over region
  • Divide integral result by region area to obtain average value

Order of integration in rectangles

  • Rectangular regions allow flexible integration order simplifying calculations
  • Identify outer and inner integrals outer determines first variable to integrate wrt
  • Consider function complexity wrt each variable choose order simplifying process
  • Fubini's Theorem justifies changing integration order ∫ab∫cdf(x,y)dydx=∫cd∫abf(x,y)dxdy\int_a^b \int_c^d f(x,y) dy dx = \int_c^d \int_a^b f(x,y) dx dy
  • Analyze function structure to determine optimal integration order (polynomial, trigonometric)

Key Terms to Review (15)

Area under surface: The area under a surface refers to the measurement of the space contained beneath a three-dimensional graph and above a specified region in the xy-plane. This concept is crucial for understanding how to calculate volumes and understand how functions behave over two dimensions by summing up infinitely small contributions across an area, leading to the application of double integrals over rectangular regions.
Average value: The average value of a function over a specific region is a measure that represents the 'typical' value of that function throughout that area. It is calculated by taking the double integral of the function over the region and dividing it by the area of that region. This concept allows for understanding how a function behaves across a two-dimensional space, making it essential for interpreting various physical and geometrical situations.
Bounded Function: A bounded function is a type of function whose output values do not exceed a certain fixed value, regardless of the input. This means there exists a real number that acts as an upper limit and another that serves as a lower limit for the values of the function. Understanding bounded functions is crucial when working with double integrals over rectangles, as these functions help ensure that the area under the curve can be accurately represented and computed.
Change of Variables Theorem: The Change of Variables Theorem is a fundamental result in multivariable calculus that allows for the transformation of integrals over one set of variables into integrals over another set. This theorem provides the necessary tools to evaluate complex multiple integrals by changing the variables of integration, often making the computation easier by simplifying the region of integration or transforming it into a more manageable shape.
Continuous Function: A continuous function is a type of function where small changes in the input result in small changes in the output. This property means that the function does not have any abrupt jumps, breaks, or holes in its graph. In the context of double integrals over rectangles, continuous functions ensure that the area under the curve can be accurately estimated, as they allow for the use of limit processes and approximation techniques that are central to evaluating integrals.
Double Integrals: Double integrals are a mathematical concept used to compute the accumulation of quantities over a two-dimensional region, effectively generalizing the idea of a single integral to multiple dimensions. They allow for the calculation of areas, volumes, and other quantities by integrating a function of two variables across a specified rectangular or more complex region. The concept is foundational in understanding how to evaluate functions over two-dimensional spaces, which connects to changing variables for more complex integration problems.
Fubini's Theorem: Fubini's Theorem is a fundamental result in calculus that provides a way to compute multiple integrals by allowing the evaluation of an integral as an iterated integral. This theorem states that if a function is continuous over a rectangular region, the double integral can be computed by iterating the integration process, first with respect to one variable and then the other. This principle also extends to triple integrals, making it crucial for changing the order of integration when dealing with more complex regions or functions.
Function of Two Variables: A function of two variables is a rule that assigns a unique output value for each ordered pair of input values from a two-dimensional domain. This means that for every pair $(x, y)$, the function produces a corresponding output $z = f(x, y)$, which can be visualized as a surface in three-dimensional space. These functions allow us to explore relationships and changes in two dimensions simultaneously, making them essential in understanding various mathematical and real-world phenomena.
Inner integral: The inner integral is the first step in a double integral, representing the integration of a function with respect to one variable while treating the other variables as constants. This process allows for the accumulation of values over a specified interval for one variable before moving on to the outer integral. Understanding the inner integral is crucial for evaluating double integrals, especially when dealing with functions defined over rectangular regions.
Iterated Integration: Iterated integration is a method used to compute double integrals by performing two successive integrations, one for each variable, over a specified region. This technique simplifies the evaluation of double integrals, especially over rectangular regions, by allowing the integral to be broken down into simpler, one-dimensional integrals. It connects closely with the geometric interpretation of area under surfaces and facilitates calculations in multivariable calculus.
Limits of Integration: Limits of integration are the values that define the range of a definite integral, specifying the interval over which a function will be integrated. In the context of double integrals over rectangles, these limits help determine the area being considered and guide the evaluation of the integral by establishing the boundaries in both dimensions. Understanding these limits is crucial for accurately calculating volumes or areas under surfaces represented by functions of two variables.
Outer Integral: The outer integral refers to the integral that is evaluated second in a double integral setup. It typically encompasses the overall integration with respect to one variable after the inner integral has been computed, allowing for the aggregation of results over a specified range. Understanding the role of the outer integral is crucial as it helps to determine how the entire area or volume is represented mathematically when integrating over two dimensions.
Region of Integration: A region of integration is a specified area over which an integral is calculated, defining the limits and boundaries for integration. It plays a critical role in determining how functions are evaluated when calculating double or triple integrals, ensuring that the area or volume being considered is accurately represented. Understanding the region of integration allows for proper setting up of integrals in both rectangular and more complex shapes, impacting the final results significantly.
Riemann Sum: A Riemann sum is a method for approximating the total area under a curve by dividing it into small rectangles, calculating the area of each rectangle, and summing those areas. This concept is foundational in understanding how to evaluate double integrals, especially over rectangular regions, by allowing us to break down complex shapes into simpler components. The accuracy of the approximation improves as the number of rectangles increases, leading to a better representation of the area being analyzed.
Volume Under Surface: Volume under a surface refers to the three-dimensional space contained beneath a given surface defined by a function of two variables over a specific region in the xy-plane. This concept is crucial in understanding how double integrals can be used to calculate the total volume enclosed between the surface and the xy-plane, providing a way to quantify physical quantities such as mass or charge distributed over a two-dimensional area.
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