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%* Navigate [[..:..:activities:link|back to the activity]].
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\Lab{The Grid}
\SecMark
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%*==== THE GRID ====
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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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Understanding different ways of expressing area using integration.
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\item
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Concrete example of Area Corollary to Green's/Stokes' Theorem.
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\end{itemize}
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}
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\noindent
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We originally used this activity after covering Green's Theorem; we now skip
Green's Theorem and do this activity shortly before Stokes' Theorem.
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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 line integrals.
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\item
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\textit{Green's Theorem is not a prerequisite!}
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%*//Green's Theorem is not a prerequisite!//
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\end{itemize}
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}
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\subsubsection{Warmup}
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%*== Warmup ==
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\begin{itemize}
\item
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The first problem is a good warmup.
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\end{itemize}
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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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a planimeter if available
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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 the magic -- finding area by walking around the boundary!
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\item
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Point out that this works for any closed curve, not just the rectangular
regions considered here.
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\item
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Demonstrate or describe a planimeter, used for instance to measure the area of
a region on a map by tracing the boundary.
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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 students use a consistent orientation on their path.
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\item
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Make sure students explicitly include all segments of their path, including
those which obviously yield zero.
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\item
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Students in a given group should all use the same curve.
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\item
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Students should be discouraged from drawing a curve whose longest side is
along a coordinate axis.
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\item
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Students may need to be reminded that $\OINT$ implies the counterclockwise
orientation. But it doesn't matter what orientation students use so long as
they are consistent!
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\item
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A geometric argument that the orientation should be reversed when
interchanging $x$ and $y$ is to rotate the $xy$-plane about the line $y=x$.
(This explains the minus sign in Green's Theorem.)
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\item
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Students may not have seen line integrals of this form (see below).
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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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Orientation of closed paths.
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\item
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Line integrals of the form $\INT P\,dx+Q\,dy$.
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\hfill\break
\textit{
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%*//
We do not discuss such integrals in class! Integrals of this form
almost always arise in applications as $\INT\FF\cdot d\rr$.
%*//
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}
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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}
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%*== Enrichment ==
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\Rich{
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\begin{itemize}
\item
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Write down Green's Theorem.
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\item
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Go to 3 dimensions --- bend the curve out of the plane and stretch the region
like a butterfly net or rubber sheet. This is the setting for Stokes'
Theorem!
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\end{itemize}
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}
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