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The Triangle: Instructor's Guide

Main Ideas

  • Multiple integrals can (in principle) be computed in any order.

Students' Task

Estimated Time: 15–30 minutes

Students evaluate an integral over a triangular region, then reverse the order of integration.

Prerequisite Knowledge

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


Activity: Introduction

A good introduction to this activity is an example of an integral over a rectangular region, evaluated using both orders of integration.

Activity: Student Conversations

Students may still be confused about “where” and “what”, expecting (single) integrals to yield area, and therefore puzzled over what the “$y$” is doing in the given integral ($\int y\,dA$).

A related confusion involves the difference between chopping a 2-dimensional region (“where”) and the chopping of the area under the graph of a function into strips (“what”!). Emphasize that the direction of integration is the direction of chopping. In the latter case, “where” corresponds to chopping up the $x$-axis (only), corresponding to the *width* of the strip; the *height* of the strip corresponds to “what”. For 2-dimensional regions, typically “what” is not shown, only “where”, and the same strips indicate the direction of integration.

Some students will use constant limits of integration; emphasize that this choice corresponds to a rectangular region. Others will chop both ways by using variable limits in both integrals; emphasize that this choice can not yield a numerical answer (and must therefore be incorrect). Some students will try simply swapping the integrals, limits and all; emphasize the importance of drawing diagrams showing the actual chopping being used.

Activity: Wrap-up

Discuss these differences between “where” and “what” with the whole class. Emphasize that both orders of integration must yield the same answer, even though the computations involved may be quite different.


Natural followups are the Triangle activity and the Matching SWBQ.

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