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Double Integrals: Instructor's Guide

Main Ideas

  • Reversing the order of integration can change the computational difficulty.

Students' Task

Estimated Time: 30–45 minutes

Students reverse the order of integration of an integral over a trianglular region, which however is given only in symbolic form.

Prerequisite Knowledge

  • Interpretation of integration as chopping, adding, and multiplying;
  • Familiarity with double integrals over rectangular regions.


Activity: Introduction

A good introduction to this activity is the Triangle activity, which can be done on the same day.

Activity: Student Conversations

Some students will use $0$ as the lower limit whenever possible, resulting in inconsistent orientations. Emphasize the need to get positive answers when integrating positive functions (such as $1$ for area), which leads to the convention that integrals over 2-dimensional regions should be evaluated from left to right, and from bottom to top.

Some students will ask whether their two integrals can be combined. (They can.)

Activity: Wrap-up

Emphasize the usefulness of drawing the region in order to determine the reversed limits. Point out that the horizontal integrals can be combined if done first, thus demonstrating that reversing the order of integration can simplify the computation.


A natural followups is the Matching SWBQ. An optional extension would be an example of an integral that can not be evaluated as given, but can after reversing the order of integration. One such example is $\int_0^6\int_{x/3}^2 x\sqrt{y^3+1}\,dy\,dx$.

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