Unit 8: Triple Integral in Rectangular Coordinate

The Concept

The definition of the double integral was introduced in Unit 5. Just as we use a double integral to integrate over a 2D region, we use a triple integral, [latex]\iiint_D f(x,y,z)\,dV[/latex], to integrate over a 3D region. Similarly, as with double integrals, the bounds of inner integrals may be functions of the outer variables. These bound functions are what encode the shape of the general region D. We may define a triple integral generally as follows:

[latex]\iiint_D f(x,y,z)\,dV=\int_{a}^{b} \int_{g_1(x)}^{g_2(x)} \int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z) \, dz dy dx[/latex]

where [latex]x=a[/latex] and [latex]x=b[/latex] represent the lower and upper bounds of [latex]x[/latex], [latex]y=g_1(x)[/latex] and [latex]y=u_2(x,y)[/latex] are the lower and upper bounds of [latex]y[/latex], and [latex]z=u_1(x,y)[/latex] and [latex]z=u_2(x,y)[/latex] are the lower and upper bounds of [latex]z[/latex]. Similar to double integrals, triple integrals are iterative as well. Thus, they can be written as different forms, such as

[latex]\iiint_D f(x,y,z)\,dV=\int_{c}^{d} \int_{h_1(y)}^{h_2(y)} \int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z) \, dz dx dy[/latex].

The Plot

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

  1. You are able to change the bounds on the triple integral in rectangular coordinates.
  2. You may input your function, [latex]f(x,y,z)[/latex], to be integrated at the bottom as well, in which the triple integral of said function will be presented at the top of the screen in the beige area.
  3. You may also change the grid size of the 3D solid depicted on the screen for the function [latex]f(x,y,z) = 1[/latex].

Self-Checking Questions

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

Evaluate the triple integrals over the rectangular solid box B.

  1. [latex]\iiint_D (2x + 3y^2 + 4z^3)\, dV[/latex], where [latex]B=\{(x,y,z)| 0 \leq x \leq 1, 0 \leq y \leq 2, 0 \leq z \leq 3\}[/latex]
  2. [latex]\iiint_D z \sin(x) + y2) \, dV[/latex], where [latex]B=\{(x,y,z)| 0 \leq x \leq \pi, 0 \leq y \leq 12, -1 \leq z \leq 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).


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3D Interactive Plots for Multivariate Calculus Copyright © 2022 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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