Calculus
Textbooks
Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Multiple Integrals
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Multiple Integrals
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 5
Created by Boundless

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)→f(ρcosφ,ρsinφ,z)f(x,y,z) \rightarrow f(\rho \cos \varphi, \rho \sin \varphi, z)f(x,y,z)→f(ρcosφ,ρsinφ,z).
    • In switching to cylindrical coordinates, the dxdydzdx\, dy\, dzdxdydz differentials in the integral become ρdρdφdz\rho \, d\rho \,d\varphi \,dzρdρdφdz.
    • Therefore, an integral evaluated in Cartesian coordinates can be switched to an integral in cylindrical coordinates as∭Df(x,y,z)dxdydz=∭Tf(ρcosφ,ρsinφ,z)ρdρdφdz\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∭​D​​f(x,y,z)dxdydz=∭​T​​f(ρcosφ,ρsinφ,z)ρdρdφdz.

Terms

  • cylindrical coordinate

    a three-dimensional 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 R3 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: 

f(x,y,z)→f(ρcosφ,ρsinφ,z)f(x,y,z) \rightarrow f(\rho \cos \varphi, \rho \sin \varphi, z)f(x,y,z)→f(ρcosφ,ρsinφ,z)

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 dxdydzdx\, dy\, dzdxdydz differentials in the integral become ρdρdφdz\rho \, d\rho \,d\varphi \,dzρdρdφdz.

Cylindrical Coordinates

Cylindrical coordinates are often used for integrations on domains with a circular base.

Example 1

The region is:

 D={x2+y2≤9, x2+y2≥4, 0≤z≤5}D = \{ x^2 + y^2 \le 9, \ x^2 + y^2 \ge 4, \ 0 \le z \le 5 \}D={x​2​​+y​2​​≤9, x​2​​+y​2​​≥4, 0≤z≤5}

If the transformation is applied, this region is obtained: 

T={2≤ρ≤3, 0≤φ≤2π, 0≤z≤5}T = \{ 2 \le \rho \le 3, \ 0 \le \varphi \le 2\pi, \ 0 \le z \le 5 \}T={2≤ρ≤3, 0≤φ≤2π, 0≤z≤5}

because the z component is unvaried during the transformation, the dxdydzdx\, dy\, dzdxdydz differentials vary as in the passage in polar coordinates: therefore, they become: ρdρdφdz\rho \, d\rho \,d\varphi \,dzρdρdφdz. Finally, it is possible to apply the final formula to cylindrical coordinates: 

∭Df(x,y,z)dxdydz=∭Tf(ρcosφ,ρsinφ,z)ρdρdφdz\displaystyle{\iiint Df(x,y,z)dx\,dy\,dz=\iiint Tf(\rho\cos\varphi,\rho\sin\varphi,z)\rho\, d\rho \,d\varphi \,dz}∭Df(x,y,z)dxdydz=∭Tf(ρcosφ,ρsinφ,z)ρdρdφdz

This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the zzz interval and even transform the circular base and the function.

Example 2

The function f(x,y,z)=x2+y2+zf(x,y,z) = x^2 + y^2 + zf(x,y,z)=x​2​​+y​2​​+z is and as integration domain this cylinder: 

D={x2+y2≤9, −5≤z≤5}D = \{ x^2 + y^2 \le 9, \ -5 \le z \le 5 \}D={x​2​​+y​2​​≤9, −5≤z≤5}

The transformation of DDD in cylindrical coordinates is the following: 

T={0≤ρ≤3, 0≤ϕ≤2π, −5≤z≤5}T = \{ 0 \le \rho \le 3, \ 0 \le \phi \le 2 \pi, \ -5 \le z \le 5 \}T={0≤ρ≤3, 0≤ϕ≤2π, −5≤z≤5}

while the function becomes:

 f(ρcosφ,ρsinφ,z)=ρ2+zf(\rho \cos \varphi, \rho \sin \varphi, z) = \rho^2 + zf(ρcosφ,ρsinφ,z)=ρ​2​​+z

Therefore, the integral becomes:

∭D(x2+y2+z)dxdydz=∭T(ρ2+z)ρdρdϕdz=∫−55dz∫02πdϕ∫03(ρ3+ρz)dρ=2π∫−55[ρ44+ρ2z2]03dz=2π∫−55(814+92z)dz=405π\begin{aligned}\displaystyle{\iiint_D (x^2 + y^2 +z) \, dx\, dy\, dz} &\displaystyle{= \iiint_T ( \rho^2 + z) \rho \, d\rho\, d\phi\, dz} \\ &= \int_{-5}^5 dz \int_0^{2 \pi} d\phi \int_0^3 ( \rho^3 + \rho z )\, d\rho\\ &= 2 \pi \int_{-5}^5 \left[ \frac{\rho^4}{4} + \frac{\rho^2 z}{2} \right]_0^3 \, dz\\ &= 2 \pi \int_{-5}^5 \left( \frac{81}{4} + \frac{9}{2} z\right)\, dz\\ &= 405 \pi \end{aligned}​∭​D​​(x​2​​+y​2​​+z)dxdydz​​​​​​​=∭​T​​(ρ​2​​+z)ρdρdϕdz​=∫​−5​5​​dz∫​0​2π​​dϕ∫​0​3​​(ρ​3​​+ρz)dρ​=2π∫​−5​5​​[​4​​ρ​4​​​​+​2​​ρ​2​​z​​]​0​3​​dz​=2π∫​−5​5​​(​4​​81​​+​2​​9​​z)dz​=405π​​

[ edit ]
Edit this content
Prev Concept
Double Integrals in Polar Coordinates
Triple Integrals in Spherical Coordinates
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.