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Triple Integrals in Cylindrical Coordinates
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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
$f(x,y,z) \rightarrow f(\rho \cos \varphi, \rho \sin \varphi, z)$ .  In switching to cylindrical coordinates, the
$dx\, dy\, dz$ differentials in the integral become$\rho \, d\rho \,d\varphi \,dz$ .  Therefore, an integral evaluated in Cartesian coordinates can be switched to an integral in cylindrical coordinates as
$\iiint_D f(x,y,z) \, dx\, dy\, dz = \iiint_T f(\rho \cos \varphi, \rho \sin \varphi, z)\rho \, d\rho \,d\varphi \,dz$ .
Terms

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

differential
an infinitesimal change in a variable, or the result of differentiation
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:
The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region. Also in switching to cylindrical coordinates, the
Cylindrical Coordinates
Cylindrical coordinates are often used for integrations on domains with a circular base.
Example 1
The region is:
If the transformation is applied, this region is obtained:
because the z component is unvaried during the transformation, the
This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the
Example 2
The function
The transformation of
while the function becomes:
Therefore, the integral becomes:
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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: Parametric Equations, Calculus with Parametric Curves, and Calculus of VectorValued Functions
 domain
 Appears in these related concepts: Introduction to Rational Functions, Restricting Domains to Find Inverses, and Composition of Functions and Decomposing a Function
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 integral
 Appears in these related concepts: Indefinite Integrals and the Net Change Theorem, The Correction Factor, and Expected Value
 integration
 Appears in these related concepts: Area and Distances, The Definite Integral, and The Substitution Rule
 polar
 Appears in these related concepts: DipoleDipole Force, Polar Coordinates, and Area and Arc Length in Polar Coordinates
 symmetry
 Appears in these related concepts: Symmetry of Functions, Balance, and Rhythm
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
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Source: Boundless. “Triple Integrals in Cylindrical Coordinates.” Boundless Calculus Boundless, 26 May. 2016. Retrieved 24 Feb. 2017 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/multipleintegrals22/tripleintegralsincylindricalcoordinates1602876/