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- 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.
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- 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)?
We originally used this activity after covering Green's Theorem; we now skip Green's Theorem and do this activity shortly before Stokes' Theorem.
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 \(\int\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}\).
Determine the area of a triangle or an ellipse by integrating along the boundary.
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?