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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.

==== Props/Equipment ====

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

==== Activity: Introduction ====

A good introduction to this activity is the
[[courses:activities:mvact:mvtriangle|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.

==== Extensions ====

A natural followups is the [[swbq:mvsw:mvswmatch|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 $\int_0^6\int_{x/3}^2 x\sqrt{y^3+1}\,dy\,dx$.