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### THE PRETZEL

#### Essentials

##### Main ideas

• Calculating (scalar) line integrals.
• Use what you know!

##### Prerequisites

• Familiarity with $d\rr$.
• Familiarity with “Use what you know” strategy.

##### Warmup

It is 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).

##### Props

• whiteboards and pens
• “linear” chocolate covered candy (e.g. Pocky)

##### Wrapup

• Emphasize that students must express each integrand in terms of a single variable prior to integration.
• Emphasize that each integral must be positive!
• Discuss several different ways of doing this problem (see below).

#### Details

##### In the Classroom

• Make sure the shape of the pretzel is clear! It might be worth drawing it on the board.
• 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.
• Polar coordinates are natural for all three parts of this problem, not just the circular arc.
• Many students will think that the integral “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.
• 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 “down” the $y$-axis at all.
• 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.
• There is of course a symmetry argument which says that the two “legs” along the axes must have the same amount of chocolate — although some students will put a minus sign into this argument!

##### Subsidiary ideas

• $ds=|d\rr|$

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