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 gives the direction that the wheel is rolling in when the planimeter
passes over the point . 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
. Since points in the direction that the wheel of the planimeter
is moving, 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
where C is the path traced
out by the planimeter?
- How could you express the quantity
represented by the integral
using an iterated integral rather
than a line integral?
- Consider the vectors shown in
the diagram below. Explain why the vector field that describes the planimeter would be expected to be a
scalar multiple of
The point -- the location of the elbow of the planimeter -- can be expressed
in terms of . You will find a command in your worksheet to solve for and
in terms of and . 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
- 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,
. 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
. 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?