Part2 of Green's Theorem and the Planimeter

Green's Theorem and the Planimeter

Part 2: Iterated Integrals and Line Integrals

In this part you will investigate relationships between iterated integrals and line integrals. The main mathematical result of this section is a theorem normally called Green's Theorem, named for the self-taught English scientist George Green (1793-1841). You will verify that a special case of this statement is true.

Theorem: Let C be a piecewise smooth, simple, closed curve in the plane, and let D be the region bounded by the curve C. The curve C is traced out in such a way that the region D is always "to the left". If the functions P(x,y) and Q(x,y) have continuous partial derivatives on an open region of the plane that contains D then:

$\begin{displaymath}\int_C P(x,y) \; dx + Q(x,y) \; dy = \int\!\!\int_D ( \frac{\... ... Q}{\partial x} - \frac{\partial P}{\partial y} ) \; dA \; . \end{displaymath}$

Several pairs of vector functions and curves are listed below. To which of these functions and curves does the theorem apply? In each case, explain your reasoning.
1. $P(x,y)=\sqrt{x^2+y^2}$ , $Q(x,y)=\sqrt{x^2+y^2}$ , C is the circle x² + y²= 1.
2. $P(x,y)=\cos(y)$ , , C is the polar coordinate curve $r^{2} = 9 \cos (\theta)$ . (Note that this polar graph is only defined on the subset of angles for which the right hand side is nonnegative.)
3. , $Q(x,y)=2x/\sqrt{x^2+y^2}$ , C is the union of the line segments joining the points (1,1) and (2,2), (2,2) and (2,3), and (2,3) and (1,1).
4. $P(x,y)=1/\sqrt{x^2+y^2}$ , $Q(x,y)=1/\sqrt{x^2+y^2}$ and C is the polar curve $r=1+\cos(\theta)$ .
Set up and evaluate the line integral
$\begin{displaymath}\int_C \cos(x) \; dx + \sin(y) \; dy \; , \end{displaymath}$
where C is the boundary of the triangular region with vertices at (0,0), (2,0) and (0,2) (see picture at right), in two ways:
1. Find a parameterization of the curve C traced out in a counter-clockwise directon, and then evaluate the line integral using this parameterization.
2. Convert the line integral into an iterated integral, and evaluate the iterated integral.
Set up and evaluate the iterated integral
$\begin{displaymath}\int\!\!\int_D (x-y) \; dA \end{displaymath}$ ,
where D is the region between the two circles x²+ y²= 1 and x² + y²= 25 (see picture at right), in two ways:
1. Evaluate an iterated integral in a convenient coordinate system.
2. Find a parameterization of the boundary curve C, and evaluate the corresponding line integral using this parameterization. (Note that the curve C has two components. Make sure that you move along each curve in such a way that the region is always "to your left".)
Based on what you have seen, list some potential uses for the theorem.

Next we examine why the theorem works in a very special case, namely, when the region is the rectangle D shown at the right, and the curve is the boundary of the rectangle, R, traced out in a counter-clockwise direction.

For several different choices of the function P(x,y), set up and evaluate the integrals
$\int_R P(x,y) \; dx$ and $\int_{a}^{b} \int_{c}^{d} -\frac{\partial P}{\partial y} \; dy dx$
Based on your results, what do you conclude about these two integrals?
For several different choices of the function , set up and evaluate the integrals
$\begin{displaymath}\int_R Q(x,y) \; dy \end{displaymath}$ and $\begin{displaymath}\int_a^b \int_c^d \frac{\partial Q}{\partial x} \; dy dx \; . \end{displaymath}$
Based on your results, what do you conclude about these two integrals?
How do steps 5 and 6 verify the truth of Green's Theorem for coordinate rectangle regions?

Do you expect the relationships that you have observed between the integrals to work for curves other than rectangles and squares? Explain.

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