Units

Unit 9: Triple Integrals in Cylindrical Coordinates

The Concept

Cylindrical coordinates are a simple extension of 2D polar coordinates to 3D. Recall that, in Unit 7, the position of a point in 2D (i.e., [latex]xy[/latex]-plane) can be described using polar coordinates [latex](r, \theta)[/latex], where [latex]r[/latex] is the distance of the point from the origin and [latex]\theta[/latex] is the angle between the [latex]x[/latex]-axis and the line segment from the origin to the point. With the addition of a third dimension, [latex]z[/latex]-axis from the Cartesian (i.e., rectangular) coordinate system, we are able to describe a point in 3D cylindrical coordinates, i.e., [latex](r, \theta, z)[/latex].

Cylindrical coordinates simply combine the polar coordinates in the [latex]xy[/latex]-plane with the usual [latex]z[/latex] coordinate of Cartesian coordinates. To form the cylindrical coordinates of a point [latex]P[/latex], simply project it down to a point [latex]Q[/latex] in the [latex]xy[/latex]-plane. Then, take the polar coordinates [latex](r, \theta)[/latex] of the point [latex]Q[/latex]. The third cylindrical coordinate is the same as the usual [latex]z[/latex]-coordinate. It is the signed distance of point [latex]P[/latex] to the [latex]xy[/latex]-plane.

The Plot

Now, you should engage with the 3D plot below to understand triple integrals in cylindrical coordinates[1]. Follow the steps below to apply changes to the plot and observe the effects:

  1. Change the bounds on the triple integral in cylindrical coordinates where and represent the outermost bounds. [latex]\alpha[/latex] and [latex]\beta[/latex] are constants and they are the lower and upper bounds of angle [latex]\theta[/latex].  [latex]r_1[/latex] and [latex]r_2[/latex] are functions of [latex]\theta[/latex]), and they are the lower and upper bounds of [latex]r[/latex].  [latex]u_1[/latex] and [latex]u_2[/latex] are functions of [latex]r[/latex] and [latex]\theta[/latex], and they are the lower and upper bounds of [latex]z[/latex].
  2. You may also change the grid size of the 3D solid depicted on the screen for the function [latex]f(x,y,z)[/latex].

Self-Checking Questions

Check your understanding by solving the following questions[2]:

Evaluate the triple integrals [latex]f(x,y,z)[/latex] over the solid [latex]E[/latex].

  1. [latex]E = {(x,y,z)| x^2+y^2 \leq 9, x \geq 0, y\geq 0, 0 \leq z \leq 1}, f(x,y,z) = z[/latex]
  2. [latex]E = {(x,y,z)| 1 \leq x^2+y^2 \leq 9,  y \geq 0, 0 \leq z \leq 1}, f(x,y,z) = x^2+y^2[/latex]

Use the graph to find the answers to these questions.


  1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.
  2. Gilbert Strang, Edwin “Jed” Herman,  OpenStax, Calculus Volume 3,  Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

3D Interactive Plots for Multivariate Calculus by Dr. Na Yu, Ryerson University is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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