# Lecture 25: Triple integrals in rectangular and cylindrical coordinates

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Lesson Summary:

In this lesson, you will learn how to compute triple integrals using both rectangular and cylindrical coordinates. The process is similar to double integrals, with one more coordinate to consider. The lesson includes examples of setting up triple integrals using different orders of integration and coordinate systems. Additionally, the concept of symmetry is discussed as a way to simplify the calculations. Overall, this lesson is a great resource for anyone who wants to improve their understanding of triple integrals in space.

Lesson Description:

Learn how to compute triple integrals using both rectangular and cylindrical coordinates.

Denis Auroux. 18.02 Multivariable Calculus, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (Accessed March 21, 2009). License: Creative Commons Attribution-Noncommercial-Share Alike.

• How do you compute a triple integral?
• What does it mean to find a triple integral?
• How do you find a triple integral using rectangular, polar, or cylindrical coordinates?
• How do you find the triple integral of 1 dV between the paraboloids z = x^2 + y^2 and z = 4 - x^2 - y^2?
• How do you know what order to write dx, dy, and dz when doing a triple integral?
• What are cylindrical coordinates?
• What are some applications of finding triple integrals?
• How do you find the moment of inertia of a 3-dimensional solid cone?
• How do you set up a triple integral for the region z > 1 - y inside the unit sphere centered at the origin, x^2 + y^2 + z^2 < 1 using rectangular coordinates?
• #### Staff Review

• Currently 4.0/5 Stars.
This lesson starts the transition from two dimensions into three dimensions. Flux, curl, work, integrals, and all other topics are simply moved into space from the plane. Triple integrals in rectangular and cylindrical coordinates are explained in this lecture. It is very important that you have a firm understanding of how double integrals work in the plane before trying to tackle this lesson. Some very involved examples are shown using rectangular, polar, and cylindrical coordinates. This is a very interesting but possibly confusing lesson.