# Lecture 16: Double integrals

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Lesson Summary:

In this lesson, you will learn about double integrals and their applications in calculus. Double integrals are used to calculate the volume below a surface in space over a specific region. The process involves cutting the region into small pieces and summing the volumes of each small box that sits under the graph, eventually taking the limit as the size of the pieces tends to zero. The integration is done iteratively, either with x or y being integrated first, and then the other variable. The lesson includes a worked example of integrating a function over a square region in the x-y plane.

Lesson Description:

Learn how to compute double integrals and what they are used for in Calculus.

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.

• What is a double integral?
• How do you find a double integral?
• What does it mean geometrically to find a double integral?
• How do you find the volume below a multivariable function?
• How do you know which order to write dx and dy in double integrals?
• What is an iterated integral?
• How do you set up a double integral?
• What is the double integral of 1 - x^2 - y^2 dydx from 0 to 1 for x and y?
• How do you find the double integral of 1 - x^2 - y^2 dydx over the quarter disc region bounded by x^2 + y^2 = 1 in the first quadrant?
• How do you find the double integral of e^y/y dydx?
• How do you switch a double integral from dydx to dxdy?
• #### Staff Review

• Currently 4.0/5 Stars.
This lesson is an introduction to double integrals of multivariable functions over regions in the plane. The professor explains very clearly the concept of what taking a double integral means, how to set up a double integral, and how to actually compute it. Some actual example problems are shown as well. This is a really good lesson on this topic. Some very difficult and confusing topics are covered.