Small Group Activity: The Grid

Vector Calculus II 2021
  • Media
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    • 2559/grid.png
  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!

Keywords
area line integrals Green's Theorem
Learning Outcomes