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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: Composition of Functions and Decomposing a Function, Visualizing Domain and Range, and Finding the Domain of a Rational Function
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 integral
 Appears in these related concepts: The Correction Factor, Integration By Parts, and Trigonometric Integrals
 integration
 Appears in these related concepts: Basic Integration Principles, Area and Distances, 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: Curve Sketching, Rhythm, and The Third Law: Symmetry in Forces
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Triple Integrals in Cylindrical Coordinates.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 23 Jul. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/multipleintegrals22/tripleintegralsincylindricalcoordinates1602876/