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Triple Integrals in Cylindrical Coordinates
When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.
Learning Objective

Evaluate triple integrals in cylindrical coordinates
Key Points
 Switching from Cartesian to cylindrical coordinates, the transformation of the function is made by the following relation
$f(x,y,z) \rightarrow f(\rho \cos \phi, \rho \sin \phi, z)$ .  In switching to cylindrical coordinates, the (dx dy dz) differentials in the integral becomes (ρ dρ dφ dz).
 Therefore, an integral evaluated in Cartesian coordinates can be switched to an integral in cylindrical coordinates as
$\iiint_D f(x,y,z) \, dx\, dy\, dz = \iiint_T f(\rho \cos \phi, \rho \sin \phi, z) \rho \, d\rho\, d\phi\, dz$ .
Terms

differential
an infinitesimal change in a variable, or the result of differentiation

cylindrical coordinate
a threedimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis
Full Text
When the function to be integrated has a cylindrical symmetry, it is sensible to change the variables into cylindrical coordinates and then perform integration.
In R^{3} the integration on domains with a circular base can be made by the passage in cylindrical coordinates ; the transformation of the function is made by the following relation:
Cylindrical Coordinates
Cylindrical coordinates are often used for integrations on domains with a circular base.
Examples
1. The region is
2. The function
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Key Term Reference
 Cartesian
 Appears in these related concepts: Cylindrical and Spherical Coordinates, Double Integrals in Polar Coordinates, and Real Numbers, Functions, and Graphs
 coordinate
 Appears in these related concepts: Hyperbolic Functions, Physics and Engeineering: Center of Mass, and Parametric Equations
 domain
 Appears in these related concepts: Restricting Domains, Phenotypic Analysis, and Classification of Prokaryotes
 function
 Appears in these related concepts: Inverse Functions, Average Value of a Function, and Functions and Their Notation
 integral
 Appears in these related concepts: Tangent and Velocity Problems, Indefinite Integrals and the Net Change Theorem, and Volumes
 integration
 Appears in these related concepts: Area and Distances, The Definite Integral, and Volumes of Revolution
 polar
 Appears in these related concepts: DipoleDipole Force, Dispersion Force, and Selective Permeability
 polar coordinate
 Appears in these related concepts: Polar Coordinates, Area and Arc Length in Polar Coordinates, and Change of Variables
 symmetry
 Appears in these related concepts: Balance, Rhythm, and The Third Law: Symmetry in Forces
 variable
 Appears in these related concepts: Fundamentals of Statistics, The Linear Function f(x) = mx + b and Slope, and Math Review
Sources
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Cite This Source
Source: Boundless. “Triple Integrals in Cylindrical Coordinates.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 12 Feb. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/multipleintegrals22/tripleintegralsincylindricalcoordinates1602876/