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Activities

Small Group Activity

30 min.

The Grid
  1. Consider the rectangle in the first quadrant of the \(xy\)-plane as in the figure with thick black lines.
    • Label the bottom horizontal edge of the rectangle \(y=c\).
    • Label the sides of the rectangle \(\Delta x\) and \(\Delta y\).
    • What is the area of the rectangle?
    • There are also 2 rectangles whose base is the \(x\)-axis, the larger of which contains both the smaller and the original rectangle. Express the area of the original rectangle as the difference between the areas of these 2 rectangles.
  2. On the grid below, draw any simple, closed, piecewise smooth curve \(C\), all of whose segments \(C_i\) are parallel either to the \(x\)-axis or to the \(y\)-axis. Your curve should not be a rectangle. Pick an origin and label it, and assume that each square is a unit square.
    • Compute the area of the region \(D\) inside \(C\) by counting the number of squares inside \(C\).
    • Evaluate the line integral \(\displaystyle \oint_C y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) by noticing that along each segment either \(x\) or \(y\) is constant, so that the integral is equal to \(\sum_{C_i} y\,\Delta x\).

      Can you relate this to Problem 1?

    • Are your answers to the preceding two calculations the same?
    • Would any of your answers change if you replaced \(y\,\boldsymbol{\hat{x}}\) by \(x\,\boldsymbol{\hat{y}}\) in part (b)?

Main ideas

  • Understanding different ways of expressing area using integration.
  • Concrete example of Area Corollary to Green's/Stokes' Theorem.

We originally used this activity after covering Green's Theorem; we now skip Green's Theorem and do this activity shortly before Stokes' Theorem.

Prerequisites

  • Familiarity with line integrals.
  • Green's Theorem is not a prerequisite!

Warmup

  • The first problem is a good warmup.

Props

  • whiteboards and pens
  • a planimeter if available

Wrapup

  • Emphasize the magic -- finding area by walking around the boundary!
  • Point out that this works for any closed curve, not just the rectangular regions considered here.
  • Demonstrate or describe a planimeter, used for instance to measure the area of a region on a map by tracing the boundary.


Details

In the Classroom

  • Make sure students use a consistent orientation on their path.
  • Make sure students explicitly include all segments of their path, including those which obviously yield zero.
  • Students in a given group should all use the same curve.
  • Students should be discouraged from drawing a curve whose longest side is along a coordinate axis.
  • 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!
  • 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.)
  • Students may not have seen line integrals of this form (see below).
  • Students do very well on this lab, particularly after working in groups for several weeks. Resist the urge to intervene.
  • Make sure everyone sees the reason \(y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) is zero on vertical pieces.
  • The issue of the negative will come up. Suggest students make a quick sketch of the vector field.
  • It is well worthwhile to do an example with a circle together as a class. The line integral should pose no trouble for them and the area of a circle is something they believe.
  • Emphasize the connection between the boundary and the interior. This is a concrete display of this relationship.

Subsidiary ideas

  • Orientation of closed paths.
  • Line integrals of the form \(\int P\,dx+Q\,dy\).

    We do not discuss such integrals in class! Integrals of this form almost always arise in applications as \(\int\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}\).

Homework

Determine the area of a triangle or an ellipse by integrating along the boundary.

Essay questions

Describe times in your life when you needed to know area (or imagine such a time). Maybe buying carpet or painting a room. What is the first step in computing area? How does this lab truly differ, if at all?

Enrichment

  • Write down Green's Theorem.
  • 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!

Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

  1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
  2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
  3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

Small Group Activity

5 min.

Acting Out Flux
Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.
  • flux electrostatics vector fields
    Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Flux Sequence sequence(s)

Small Group Activity

30 min.

Using \(pV\) and \(TS\) Plots
  • Work as area under curve in a \(pV\) plot
  • Heat transfer as area under a curve in a \(TS\) plot
  • Reminder that internal energy is a state function
  • Reminder of First Law

Small Group Activity

30 min.

Vector Surface and Volume Elements

Students use known algebraic expressions for vector line elements \(d\boldsymbol{\vec{r}}\) to determine all simple vector area \(d\boldsymbol{\vec{A}}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

  • Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small Group Activity

30 min.

Scalar Surface and Volume Elements

Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Whole Class Activity

10 min.

Pineapples and Pumpkins

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area and volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distributed to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

  • Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small Group Activity

10 min.

Cross Product
This small group activity is designed to help students visualize the cross product. Students work in small groups to determine the area of a triangle in space. The whole class wrap-up discussion emphasizes the geometric interpretation of the cross product.

Small Group Activity

30 min.

Visualization of Curl
  • A component of the curl of a vector field (at a point) is the circulation per unit area around an infinitesimal loop.
  • How to predict the sign and relative magnitude of the curl from graphs of a vector field.
  • (Optional) How to calculate the curl of a vector field using computer algebra.
  • Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence sequence(s)