Student handout: The Grid
Vector Calculus II 2021
- group Small Group Activity 30 min. planimeter if available Student handout (PDF)
- Search for related topics
area line integrals Green's Theorem
- 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.
- 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)?
- Keywords
- area line integrals Green's Theorem
- Learning Outcomes
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