[BUG] learning is not linear

[Smith, Alexander J.]

Martin,

I know what you mean!

I am reminded how every semester when we get to double and triple integrals, the strongest students start emailing me when their integrals on homework end up being zero. It happens all over again when we get to flux integrals, etc. I have an archive of these emails filled with comments like "an integral can't be zero since it is an area." Keep in mind that it is the >best< students who are alarmed at getting 0 when they do an integral. [Maybe it is a deep seated psychological issue related to a nightmare of getting 0 on an exam?]


I have learned how to get fun rapport with these students when this happens. I build on the Rodney Dangerfield tact, except it is "0 never gets any respect." I kid them that they probably got stressed in an earlier calculus course when their definite integral ended up being 0 or -, but they eventually got over it, or did they? 

Alex

..........


Tevian's comments on signs for multiple integrals struck home last
evening when I was grading an assignment.  The specific problem
involving integrating a given function over "the region inside the
petal of the polar curve r=sin(4*theta) that is in the third quadrant
and closest to the negative y-axis."  The integrand has negative
values for this region so one expects the integral to be negative.  I
got a positive result in my first attempt!  For students, one
subtlety is that the petal in question is traced out for theta
ranging from pi/4 to pi/2.  I set up a polar description of the
region with pi/4 < theta < pi/2 and sin(4*theta) < r < 0.  This
choice of bounds on r gives dr>0.  When I computed the integral, I
got a positive result because I used dA=r*dr*dtheta.  I should have
used dA=|r*dr*dtheta|=-r*dr*dtheta with the factor of -1 coming in
from |r|=-r since r<0 in the interval relevant interval.  It took me
a few minutes to find my mistake because I was focused on the sign of
dr and forgot all about the factor of r in dA.

Ironically, most students got the correct result because they used 0
for the lower limit of integration and sin(4*theta) for the upper
limit of integration This gives dr<0 and thus the extra factor of -1
needed to compensate.

Martin




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>From: Tevian Dray <tevian at math.oregonstate.edu>
>Date: Sun, 06 Feb 2005 22:40:06 -0800
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>Subject: [BUG] multiple integration
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>I'm teaching multiple integration for the first time in several years.
>I'm emphasizing the idea that integrals are sums, among other things by the
>use of questions like:
>       Is the integral of y^3 dA positive, negative, or zero
>       over a region in the plane where y is negative?
>Two unexpected (to me) issues have arisen, one of which I feel is simply an
>error by the students, but the other is more subtle.
>
>The first issue is the desire of many students to integrate before answering
>the question, obtaining y^4/4, and arguing that this is positive, so the
>integral must be positive.  This is simply a mistake, but it does indicate how
>hard it is to get students to think about what integrals mean without
>calculating anything.
>
>The second issue is the subtle assumption that dA itself is positive.  This
>seems eminently reasonable when it really is an area, but is less obvious in
>contexts with different units.  More importantly, issues of orientation are
>being swept under the rug here.  One student explicitly integrated over all
>negative values of y, but integrated from 0 to minus infinity rather than the
>other way around.  Again, this is a mistake, but where is this taught?  The
>only good explanation I know involves surface integrals, where dA is the
>magnitude of the vector surface element, and hence contains an absolute value.
>
>Most texts hide this in the "definition" of multiple integrals as iterated
>integrals, which has an orientation built in.  The instructor can emphasize
>that the limits of such integrals always go from small values to large ones,
>but that can cause problems later when dealing with surface integrals, where
>there is no reason to make this assumption.  In fact, I think it's a bad idea
>to argue, as some mathematicians do, that one should always choose
>parameterizations of curves and surfaces so that the parameters increase;
>better to teach students to integrate from "start" to "finish" regardless of
>what that implies for the parameters.  But how then should one introduce this
>concept in multiple integration?
>
>I think the answer must be to emphasize that the area element dA really has an
>absolute value in it.  Whether this is done at the level of Riemann sums, as
>       delta A = |delta x| |delta y|
>or in differential notation simply as
>       dA = |dx| |dy|
>or simply in words doesn't really matter, so long as dA is explicitly assumed
>to be positive.
>
>Comments welcome.
>
>Tevian
>
>
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