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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:
Examples
1.
The region is
2. The function
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Key Term Reference
 Cartesian
 Appears in this related concepts: Cylindrical and Spherical Coordinates, Double Integrals in Polar Coordinates, and Real Numbers, Functions, and Graphs
 coordinate
 Appears in this related concepts: Parametric Equations, Calculus with Parametric Curves, and Calculus of VectorValued Functions
 domain
 Appears in this related concepts: VectorValued Functions, Phenotypic Analysis, and Classification of Prokaryotes
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Average Value of a Function
 integral
 Appears in this related concepts: The Correction Factor, Integration Using Tables and Computers, and Volumes
 integration
 Appears in this related concepts: Growth Strategy, The Definite Integral, and The Substitution Rule
 polar
 Appears in this related concepts: DipoleDipole Force, Polar Coordinates, and Selective Permeability
 polar coordinate
 Appears in this related concepts: Area and Arc Length in Polar Coordinates, Conic Sections in Polar Coordinates, and Change of Variables
 symmetry
 Appears in this related concepts: Curve Sketching, Balance, and Rhythm
 variable
 Appears in this related concepts: Related Rates, Fundamentals of Statistics, and Math Review
Sources
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Cite This Source
Source: Boundless. “Triple Integrals in Cylindrical Coordinates.” Boundless Calculus. Boundless, 14 Nov. 2014. Retrieved 21 May. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/multipleintegrals22/tripleintegralsincylindricalcoordinates1602876/