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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.

==== Props/Equipment ====

  * [[:Props:start#whiteboards|Tabletop Whiteboard]] with markers
  * A handout for each student

==== 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.

==== Extensions ====

Natural followups are the [[courses:activities:mvact:mvtriangle|Triangle]]
activity and the [[swbq:mvsw:mvswmatch|Matching]] SWBQ.