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\Lab{The Pretzel}
\SecMark
\label{pretzel}
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%*==== THE PRETZEL ====
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\subsection{Essentials}
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%*=== Essentials ===
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\subsubsection{Main ideas}
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%*== Main ideas ==
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\Goal{
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\begin{itemize}
\item
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Calculating (scalar) line integrals.
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\item
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Use what you know!
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\end{itemize}
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}
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\subsubsection{Prerequisites}
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%*== Prerequisites ==
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\Req{
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\begin{itemize}
\item
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Familiarity with $d\rr$.
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\item
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Familiarity with
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``Use what you know''
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%*"Use what you know"
strategy.
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\end{itemize}
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}
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\subsubsection{Warmup}
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%*== Warmup ==
It is
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\textit{not}
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%*//not//
necessary to explicitly introduce scalar line integrals,
before this lab; figuring out that the (scalar) line element must be
$|d\rr|$ can be made part of the activity (if time permits).
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\subsubsection{Props}
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%*== Props ==
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\begin{itemize}
\item
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whiteboards and pens
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\item
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``linear''
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%*"linear"
chocolate covered candy (e.g. Pocky)
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\end{itemize}
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\subsubsection{Wrapup}
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%*== Wrapup ==
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\begin{itemize}
\item
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Emphasize that students must express each integrand in terms of a single
variable prior to integration.
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\item
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Emphasize that each integral must be positive!
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\item
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Discuss several different ways of doing this problem (see below).
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\end{itemize}
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\newpage
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\subsection{Details}
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%*=== Details ===
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\subsubsection{In the Classroom}
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%*== In the Classroom ==
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\begin{itemize}
\item
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Make sure the shape of the pretzel is clear! It might be worth drawing it on
the board.
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\item
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Some students will work geometrically, determining $ds$ on each piece by
inspection. This is fine, but encourage such students to try using $d\rr$
afterwards.
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\item
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Polar coordinates are natural for all three parts of this problem, not just
the circular arc.
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\item
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Many students will think that the integral
%/*
``down''
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%*"down"
the $y$-axis should be negative. They will argue that $ds=dy$, but the limits
are from $2$ to $0$. The resolution is that $ds=|dy\,\ii|=|dy|=-dy$ when
integrating in this direction.
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\item
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Unlike work or circulation, the amount of chocolate does not depend on which
way one integrates, so there is in fact no need to integrate
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``down''
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%*"down"
the $y$-axis at all.
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\item
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%*-
Some students may argue that $d\rr=\TT\,ds\Longrightarrow ds=d\rr\cdot\TT$,
and use this to get the signs right. This is fine if it comes up, but the
unit tangent vector $\TT$ is not a fundamental part of our approach.
%*
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\item
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There is of course a symmetry argument which says that the two
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``legs''
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%*"legs"
along the axes must have the same amount of chocolate --- although some
students will put a minus sign into this argument!
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\item
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\end{itemize}
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\subsubsection{Subsidiary ideas}
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%*== Subsidiary ideas ==
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\Sub{
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\begin{itemize}
\item
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\end{itemize}
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}
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\subsubsection{Homework \None}
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%*== Homework ==
%* (none yet)
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\HW{
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}
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\subsubsection{Essay questions \None}
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%*== Essay questions ==
%* (none yet)
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\Essay{
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}
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\subsubsection{Enrichment \None}
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%*== Enrichment ==
%* (none yet)
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\Rich{
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}
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