5.3 Double Integrals in Polar Coordinates

3 min read•june 24, 2024

offer a powerful alternative to rectangular coordinates when working with double integrals. They're especially useful for circular or radially symmetric regions, simplifying calculations that would be complex in rectangular form.

To use polar coordinates, we convert (x, y) to (r, θ) and adjust our integrals accordingly. This involves changing the integrand, switching dx dy to r dr dθ, and determining new integration limits. It's a game-changer for certain problems!

Polar Coordinates in Double Integrals

Rectangular to polar coordinate conversion

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  • Convert rectangular coordinates (x,y)(x, y) to polar coordinates (r,θ)(r, \theta) using x=rcos⁡θx = r \cos \theta and y=rsin⁡θy = r \sin \theta
  • Substitute dxdydx dy with rdrdθr dr d\theta in the
  • Express the integrand function f(x,y)f(x, y) in terms of rr and θ\theta using the conversion formulas
  • Determine the limits of integration for rr based on the curves bounding the region (rr limits represent the distance from the origin, also known as the )
  • Determine the limits of integration for θ\theta based on the angles formed by the bounding curves (θ\theta limits represent the angular range, also known as the )

Double integrals in polar coordinates

  • Set up the double integral in polar coordinates using the limits of integration for rr and θ\theta determined from the given region
  • Rewrite the integrand function f(x,y)f(x, y) as f(r,θ)f(r, \theta) using the polar coordinate conversion formulas
  • Integrate with respect to rr first, treating θ\theta as a constant
  • Integrate the resulting expression with respect to θ\theta
  • Apply trigonometric identities and integral properties to simplify the integration process (sum, difference, double angle formulas)
  • Evaluate the integral and simplify the final result

Applications of polar double integrals

  • Calculate the area of a region in polar coordinates using the formula A=∬DrdrdθA = \iint_D r dr d\theta, where DD is the region bounded by the given curves
    • Set up the double integral with the appropriate limits of integration for rr and θ\theta
    • Evaluate the integral to find the area
  • Find the volume of a solid with a polar base and a given height function h(r,θ)h(r, \theta) using the formula V=∬Dh(r,θ)rdrdθV = \iint_D h(r, \theta) r dr d\theta, where DD is the region in the xyxy-plane representing the base
    • Express the height function h(r,θ)h(r, \theta) in terms of rr and θ\theta
    • Set up the double integral with the appropriate limits of integration for rr and θ\theta
    • Evaluate the integral to find the volume

Limits of integration for polar regions

  • Interpret the geometric meaning of the limits of integration in polar coordinates
  • For a region bounded by curves r=f(θ)r = f(\theta) and r=g(θ)r = g(\theta), with θ\theta ranging from α\alpha to β\beta:
    • The rr limits are from the inner curve f(θ)f(\theta) to the outer curve g(θ)g(\theta) (representing the distance from the origin)
    • The θ\theta limits are from the starting angle α\alpha to the ending angle β\beta (representing the angular range)
  • For a region bounded by lines θ=α\theta = \alpha and θ=β\theta = \beta, with rr ranging from aa to bb:
    • The rr limits are from the inner radius aa to the outer radius bb (representing the distance from the origin)
    • The θ\theta limits are from the starting angle α\alpha to the ending angle β\beta (representing the angular range)
  • Sketch the region in the xyxy-plane to visualize the boundaries and determine the appropriate limits of integration (sectors, annuli)

Polar curves and forms

  • Understand how equations in (r as a function of θ) describe polar curves
  • Recognize common polar curves such as cardioids, limaçons, and roses
  • Identify the shape of a by analyzing its equation
  • Use polar curves as boundaries when setting up double integrals in polar coordinates

Key Terms to Review (15)

Angular coordinate: An angular coordinate is a value that specifies the angle at which a point is located in a polar coordinate system. This angle is measured from a reference direction, usually the positive x-axis, and helps determine the position of points in a two-dimensional plane when expressed in terms of their distance from the origin and their direction. It plays a crucial role in converting between polar and Cartesian coordinates and is essential for calculating areas, arc lengths, and performing double integrals in polar coordinates.
Annulus: An annulus is a two-dimensional shape defined as the region between two concentric circles with different radii. This geometric figure plays a significant role in various applications, particularly when dealing with areas and integration in polar coordinates, allowing for simplified calculations and analysis of circular regions.
Change of Variables: Change of variables is a mathematical technique used to transform an integral from one set of variables to another. This transformation allows for simplification and evaluation of integrals that would otherwise be difficult or impossible to solve in their original form.
Circular region: A circular region is a two-dimensional area defined by all points that are at a fixed distance, known as the radius, from a central point, called the center. This geometric shape is fundamental in mathematics and plays a crucial role in calculus, especially when working with double integrals in polar coordinates. Understanding circular regions helps in simplifying calculations and visualizing areas bounded by curves in the coordinate system.
Double Integral: A double integral is a type of multiple integral used to calculate a quantity over a two-dimensional region. It represents the integration of a function with respect to two independent variables, often denoted as $dx$ and $dy$, over a specified area or domain.
Fubini's theorem: Fubini's theorem is a fundamental principle in calculus that allows the evaluation of double integrals by iteratively integrating with respect to one variable at a time. This theorem establishes that if a function is continuous on a rectangular region, then the double integral can be computed as an iterated integral, making it possible to switch the order of integration without changing the value of the integral.
Jacobian: The Jacobian is a matrix that represents the rates of change of a vector-valued function with respect to its variables, capturing how the output changes as the input varies. This concept is crucial for transformations in multiple integrals, especially when changing from one coordinate system to another, ensuring accurate calculation of area and volume.
Polar Coordinates: Polar coordinates are a two-dimensional coordinate system that specifies the location of a point by using a distance from a fixed reference point, and an angle measured from a fixed reference direction. This system provides an alternative to the more commonly used Cartesian coordinate system, which uses perpendicular x and y axes.
Polar Curve: A polar curve, also known as a polar graph, is a graphical representation of a function in polar coordinates. It is a curve that is defined by the relationship between the radial distance (r) and the angular position (θ) of a point in a polar coordinate system.
Polar Equation: A polar equation is a mathematical representation that describes the relationship between the radius and angle in a polar coordinate system. This system uses the angle measured from a reference direction and the distance from a fixed point (the pole) to define the position of points in a plane. Polar equations often allow for easier analysis and graphing of curves that are more complex in Cartesian coordinates.
Polar Form: Polar form is a way of representing a point or a function in the Cartesian coordinate system using polar coordinates instead of rectangular coordinates. It involves specifying a point or function in terms of a distance from the origin (the radius or magnitude) and the angle between the positive x-axis and the line connecting the origin to the point (the angle or argument).
Polar Rectangle: A polar rectangle is a region in the polar coordinate system that is bounded by two radial lines and two circular arcs. It is a fundamental shape used in the evaluation of double integrals in polar coordinates.
R dθ dr: The term 'r dθ dr' represents an infinitesimal element of area in polar coordinates. It is a fundamental component in the evaluation of double integrals expressed in polar form, as it describes the infinitesimal change in the radial and angular directions within the integration domain.
Radial Coordinate: The radial coordinate is a fundamental component of polar, cylindrical, and spherical coordinate systems. It represents the distance from a fixed origin point to a specific point in space, measured along a straight line.
Sector: A sector is a specific region or portion of a circular or polar coordinate system, defined by two radial lines and the arc between them. Sectors are fundamental concepts in the study of area and arc length in polar coordinates, as well as in the evaluation of double integrals in polar coordinates.
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