Fwd: [BUG] multiple integration

[Martin Jackson]

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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