Part 5 of Green's Theorem and the Planimeter

Green's Theorem and the Planimeter

Part 5: A Mathematical Model for the Planimeter

Next you will analyze the planimeter using Green's Theorem from Part 2, which linked line integrals and iterated integrals over regions of the plane.

To create a mathematical model of the planimeter, you need to describe how the wheel on the moving end of the planimeter turns. This is because all the planimeter records is the number of rotations that the wheel has completed. One way to represent this is to define a vector field in which the vector located at a point $(x,y)$ gives the direction that the wheel is rolling in when the planimeter passes over the point $(x,y)$ . We show part of this vector field in the following figure.

The applet below shows the planimeter and the part of the vector field that is nearest the spinning wheel of the planimeter.

Your worksheet gives the commands to generate graphical representations of several different vector fields. Which of these vector fields would be the most suitable for describing the planimeter?
Call the vector field that you have selected $\vec{F}(x,y) = P(x,y)\vec{i} +Q(x,y)\vec{j}$ . Since $\vec{F}(x,y)$ points in the direction that the wheel of the planimeter is moving, $\vec{F}(x,y)$ gives a scalar multiple of the velocity vector of the planimeter's wheel. In terms of the planimeter, what interpretation would you assign to the line integral

$\begin{displaymath}\int_{C} \vec{F} \cdot d\vec{r} = \int_{C} P(x,y) \; dx + Q(x,y) \; dy \;, \end{displaymath}$

where C is the path traced out by the planimeter?
How could you express the quantity represented by the integral $\int_{C} P(x,y) \; dx + Q(x,y) \; dy$ using an iterated integral rather than a line integral?
Consider the vectors shown in the diagram below. Explain why the vector field $\vec{F}(x,y)$ that describes the planimeter would be expected to be a scalar multiple of

$\begin{displaymath}\vec{F} (x,y) = (-y+b)\vec{i} + (x-a)\vec{j} \; . \end{displaymath}$

The point $(a,b)$ -- the location of the elbow of the planimeter -- can be expressed in terms of $(x,y)$ . You will find a command in your worksheet to solve for and in terms of and $y$ . The method is to realize that if the two "arms" of the planimeter are the same length (say length 1), then the distance formula gives

$\begin{eqnarray*} (a-0)^2 + (b-0)^2 &=& 1 \\ (a-x)^2 + (b-y)^2 &=& 1 \; . \end{eqnarray*}$

Use your computer algebra system to solve for and in terms of and . Use these results to find a symbolic representation for the planimeter vector field, $\vec{F}(x,y) = P(x,y)\vec{i} +Q(x,y)\vec{j}$ . How can you check to make sure that the vector field you have defined is the right vector field for describing the planimeter?
Consider the quantity that you plan to integrate in the iterated integral that you found to be equal to the line integral $\int_{C} P(x,y)dx + Q(x,y)dy$ . Use your computer algebra system to evaluate and simplify that expression as much as you possibly can.
Suppose that the planimeter was traced around a region D of the plane. What property of the region D would your iterated integral be equal or closely related to?
Look back at the conversion formula that you made for the planimeter applet in the previous part of the module. How can you account for the algebraic structure of your conversion formula in terms of the mathematics that you have just worked out?

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