Small Group Activity: The Pretzel

Vector Calculus II 2023
A pretzel is to be dipped in chocolate. The pretzel is in the shape of a quarter circle, consisting of a straight segment from the origin to the point (2,0), a circular arc from there to (0,2), followed by a straight segment back to the origin; all distances are in centimeters. The (linear) density of chocolate on the pretzel is given by \(\lambda = 3(x^ 2 + y^2 )\) in grams per centimeter. Find the total amount of chocolate on the pretzel.

Main ideas

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

Prerequisites

  • Familiarity with \(d\boldsymbol{\vec{r}}\).
  • 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 \(|d\boldsymbol{\vec{r}}|\) 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 \(d\vec{r}\) 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 \(ds = |dy\,\boldsymbol{\hat x}|=|dy|=-dy\) 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 \(d\boldsymbol{\vec{r}}=\boldsymbol{\hat T}\,ds\Longrightarrow ds=d\boldsymbol{\vec{r}}\cdot\boldsymbol{\hat T}\), and use this to get the signs right. This is fine if it comes up, but the unit tangent vector \(\boldsymbol{\hat T}\) 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

  • \(ds=|d\boldsymbol{\vec{r}}|\)
Keywords
scalar line integrals density chocolate!
Learning Outcomes