5.5 Triple Integrals in Cylindrical and Spherical Coordinates

4 min read•june 24, 2024

Cylindrical and are powerful tools for simplifying complex three-dimensional problems. They transform rectangular coordinates into more intuitive systems, making it easier to describe and analyze objects with circular or spherical .

These coordinate systems shine when dealing with cylinders, spheres, and other symmetrical shapes. By converting between coordinate systems and using the appropriate volume elements, we can tackle integrals that would be much harder in rectangular coordinates.

Cylindrical Coordinates

Coordinate system conversions

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  • ([r](https://www.fiveableKeyTerm:r),θ,[z](https://www.fiveableKeyTerm:z))([r](https://www.fiveableKeyTerm:r), \theta, [z](https://www.fiveableKeyTerm:z)) convert to rectangular coordinates (x,y,z)(x, y, z) using trigonometric functions
    • x=rcos⁡θx = r \cos \theta represents the x-coordinate as the product of the radial distance rr and the cosine of the angle θ\theta (polar angle)
    • y=rsin⁡θy = r \sin \theta represents the y-coordinate as the product of the radial distance rr and the sine of the angle θ\theta
    • z=zz = z remains unchanged as the vertical coordinate
  • The in cylindrical coordinates [dV](https://www.fiveableKeyTerm:dV)=r dr dθ dz[dV](https://www.fiveableKeyTerm:dV) = r \, dr \, d\theta \, dz accounts for the change in volume when integrating
    • rr represents the determinant, which is necessary for the
  • Converting a from rectangular to cylindrical coordinates involves substitution and adjusting the limits of integration
    • Replace xx with rcos⁡θr \cos \theta and yy with rsin⁡θr \sin \theta in the integrand
    • Replace dx dy dzdx \, dy \, dz with r dr dθ dzr \, dr \, d\theta \, dz to account for the change in volume element
    • Determine the new limits of integration for rr, θ\theta, and zz based on the given region (, disk)

Cylindrical coordinates for symmetrical solids

  • Cylindrical coordinates simplify triple integrals for regions with circular or cylindrical symmetry (cylinders, disks, rings)
  • Evaluating a triple integral in cylindrical coordinates involves setting up the integral with the appropriate limits and volume element
    1. Identify the limits of integration for rr (radial distance), θ\theta (polar angle), and zz (vertical coordinate)
    2. Set up the integral using the volume element r dr dθ dzr \, dr \, d\theta \, dz
    3. Integrate with respect to zz, then θ\theta, and finally rr, applying the limits of integration
  • Common regions described using cylindrical coordinates include:
    • Cylinder: 0≤r≤a0 \leq r \leq a, 0≤θ≤2π0 \leq \theta \leq 2\pi, c≤z≤dc \leq z \leq d (circular base, constant height)
    • Disk: 0≤r≤a0 \leq r \leq a, 0≤θ≤2π0 \leq \theta \leq 2\pi, z=cz = c (circular region in a plane)
    • : formed by rotating a planar region around an axis

Spherical Coordinates

Coordinate system conversions

  • Spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi) convert to rectangular coordinates (x,y,z)(x, y, z) using trigonometric functions
    • x=ρsin⁡ϕcos⁡θx = \rho \sin \phi \cos \theta represents the x-coordinate using the radial distance ρ\rho, polar angle θ\theta, and azimuthal angle ϕ\phi
    • y=ρsin⁡ϕsin⁡θy = \rho \sin \phi \sin \theta represents the y-coordinate using the radial distance ρ\rho, polar angle θ\theta, and azimuthal angle ϕ\phi
    • z=ρcos⁡ϕz = \rho \cos \phi represents the z-coordinate using the radial distance ρ\rho and azimuthal angle ϕ\phi
  • The volume element in spherical coordinates dV=ρ2sin⁡ϕ dρ dθ dϕdV = \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi accounts for the change in volume when integrating
    • ρ2sin⁡ϕ\rho^2 \sin \phi represents the Jacobian determinant, which is necessary for the change of variables
  • Converting a triple integral from rectangular to spherical coordinates involves substitution and adjusting the limits of integration
    • Replace xx, yy, and zz with their spherical coordinate equivalents in the integrand
    • Replace dx dy dzdx \, dy \, dz with ρ2sin⁡ϕ dρ dθ dϕ\rho^2 \sin \phi \, d\rho \, d\theta \, d\phi to account for the change in volume element
    • Determine the new limits of integration for ρ\rho, θ\theta, and ϕ\phi based on the given region (, spherical shell)

Spherical coordinates for spherical regions

  • Spherical coordinates simplify triple integrals for regions with spherical symmetry (spheres, spherical shells, spherical wedges)
  • Evaluating a triple integral in spherical coordinates involves setting up the integral with the appropriate limits and volume element
    1. Identify the limits of integration for ρ\rho (radial distance), θ\theta (polar angle), and ϕ\phi (azimuthal angle)
    2. Set up the integral using the volume element ρ2sin⁡ϕ dρ dθ dϕ\rho^2 \sin \phi \, d\rho \, d\theta \, d\phi
    3. Integrate with respect to ϕ\phi, then θ\theta, and finally ρ\rho, applying the limits of integration
  • Common regions described using spherical coordinates include:
    • Sphere: 0≤ρ≤a0 \leq \rho \leq a, 0≤θ≤2π0 \leq \theta \leq 2\pi, 0≤ϕ≤π0 \leq \phi \leq \pi (solid sphere with radius aa)
    • Spherical shell: a≤ρ≤ba \leq \rho \leq b, 0≤θ≤2π0 \leq \theta \leq 2\pi, 0≤ϕ≤π0 \leq \phi \leq \pi (hollow sphere with inner radius aa and outer radius bb)
    • Spherical wedge: 0≤ρ≤a0 \leq \rho \leq a, α≤θ≤β\alpha \leq \theta \leq \beta, γ≤ϕ≤δ\gamma \leq \phi \leq \delta (portion of a sphere bounded by angles)

Integration Considerations

  • Symmetry: Utilize the symmetry of the region to simplify integration when possible
  • : Choose the order of integration that simplifies the problem most effectively
  • Volume element: Ensure the correct volume element is used for the chosen coordinate system

Key Terms to Review (21)

Change of Variables Theorem: The Change of Variables Theorem is a fundamental principle in calculus that allows the evaluation of integrals by transforming them into a different coordinate system. This theorem is particularly useful when dealing with complex integrals, as it simplifies the process by changing the variables involved to ones that better suit the region of integration, making calculations more manageable.
Cone: A cone is a three-dimensional geometric shape that has a circular base and tapers to a single point, called the vertex. Cones are fundamental shapes in mathematics and have important applications in various fields, including calculus and vector analysis.
Coordinate Transformation: Coordinate transformation is the process of changing the coordinate system used to represent a geometric object or a mathematical function. It involves mapping the coordinates of a point from one coordinate system to another, allowing for the analysis and manipulation of data in a more convenient or meaningful way.
Cylinder: A cylinder is a three-dimensional geometric shape that is formed by the rotation of a rectangle around one of its sides. It has a circular base and a curved surface that connects the two parallel circular bases.
Cylindrical Coordinates: Cylindrical coordinates are an alternative coordinate system used to describe the position of a point in three-dimensional space. Unlike the traditional Cartesian coordinate system, cylindrical coordinates use a radial distance, an angle, and a height to uniquely identify a location, providing a more natural way to represent certain geometric shapes and physical phenomena.
DV: In calculus, 'dV' represents an infinitesimal volume element used in the context of triple integrals. It helps to break down three-dimensional regions into smaller, manageable parts for integration. This concept is essential when calculating volumes or integrating functions over a three-dimensional space, and it can take different forms depending on the coordinate system being used, such as Cartesian, cylindrical, or spherical coordinates.
Integration Order: Integration order refers to the sequence in which multiple integrals are evaluated when dealing with multidimensional integration, such as triple integrals in cylindrical and spherical coordinates. The order of integration determines the structure of the integral and the way the integration is carried out.
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.
R: The variable 'r' represents the radial distance or the distance from the origin in polar, cylindrical, and spherical coordinate systems. It is a fundamental component that describes the location of a point in these coordinate systems, which are widely used in various areas of mathematics and physics.
R dθ dz dr: The term 'r dθ dz dr' is a key component in the context of triple integrals expressed in cylindrical and spherical coordinate systems. It represents the infinitesimal volume element that is used to integrate a function over a three-dimensional region in these coordinate systems.
Rho (ρ): Rho (ρ) is a Greek letter used to represent a variable or parameter in various mathematical and scientific contexts. In the fields of calculus and vector analysis, ρ is a crucial variable that is often used to describe the radial distance or radius in cylindrical and spherical coordinate systems.
Solid of Revolution: A solid of revolution is a three-dimensional shape formed by rotating a two-dimensional shape around an axis. This concept is key when calculating volumes and surface areas using triple integrals in cylindrical and spherical coordinates, as it allows us to simplify complex shapes into forms that are easier to analyze mathematically.
Sphere: A sphere is a three-dimensional geometric shape that is perfectly round, with all points on its surface equidistant from the center. It is one of the most fundamental shapes in mathematics and has numerous applications in various fields, including calculus, physics, and engineering.
Spherical Coordinates: Spherical coordinates are a three-dimensional coordinate system that uses three values, $r$, $\theta$, and $\phi$, to specify the location of a point in space. This system provides a natural way to describe positions on the surface of a sphere or within a spherical volume, and is widely used in various fields of mathematics, physics, and engineering.
Symmetry: Symmetry refers to the property of an object or function that remains unchanged under certain transformations, such as rotation, reflection, or translation. It is a fundamental concept in mathematics and physics that describes the inherent balance and regularity of a system.
Theta (θ): Theta (θ) is an angular coordinate that represents the position of a point in a polar, cylindrical, or spherical coordinate system. It is the angle measured counterclockwise from a reference direction, typically the positive x-axis in the xy-plane or the positive z-axis in three-dimensional space.
Triple Integral: A triple integral is a three-dimensional extension of the definite integral, used to calculate the volume of a three-dimensional region or to integrate a function over a three-dimensional domain. It is a fundamental concept in multivariable calculus and is closely related to the understanding of limits, continuity, and changes of variables in multiple integrals.
Volume Element: The volume element, also known as the differential volume, is a fundamental concept in multiple integral calculus that represents an infinitesimally small volume within a larger three-dimensional region. This volume element is a crucial component in the evaluation of triple integrals, as well as in the transformation of integrals from one coordinate system to another, such as cylindrical or spherical coordinates.
Z: In the context of three-dimensional coordinate systems, 'z' represents the vertical axis, typically used to denote height or depth. This coordinate is crucial for distinguishing points in space, allowing for the representation of three-dimensional figures and facilitating calculations involving volume and area. Understanding 'z' helps in converting between different coordinate systems, such as cylindrical and spherical coordinates.
ρ² sin φ dρ dφ dθ: The expression ρ² sin φ dρ dφ dθ represents the differential volume element in spherical coordinates. This term is crucial when setting up triple integrals in spherical coordinates as it accounts for the change in volume as we move in three-dimensional space. Understanding this term helps visualize how spherical coordinates relate to Cartesian coordinates and how to properly calculate volumes and integrals in different coordinate systems.
φ: Phi (φ) is a mathematical symbol that represents an angle, specifically the angle between a vector and a reference axis. It is a fundamental term in the context of cylindrical and spherical coordinate systems, as well as in the evaluation of triple integrals using these coordinate systems.
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