Mathematics 162                              Laboratory 13                        Week of April 26, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994

Polar Coordinates

In today's laboratory, you will become familiar with plotting in polar coordinates, and compute the areas of some regions which are bounded by the graphs of polar equations.  Before we begin the lab, enter the following command to load some special commands we will use:

<<Graphics`Graphics`; <<Graphics`Colors`

Part 1: Plotting in Polar Coordinates

As we learned in lecture, parametic equations of the form x = r(θ) cosθ, y = r(θ) sinθ can be thought of more simply in polar coordinates.  For instance, consider these equations with r(θ) = π/θ:

r[theta_] := Pi/theta;

x[theta_] := r[theta] Cos[theta];
y[theta_] := r[theta] Sin[theta]

We can plot these using the ParametricPlot command just as we learned in the last two labs:

oneplot = ParametricPlot[{x[t],y[t]},{t,Pi/6,5Pi/2},
             PlotStyle->Red, AspectRatio->Automatic]

This graph can be described in polar coordinates by r = π/θ, and Mathematica can graph it from this equation using the PolarPlot command.

otherplot = PolarPlot[Pi/t, {t, Pi/6, 5Pi/2},
          PlotStyle->Blue]

We can see that these are really the same graph by entering

Show[oneplot, otherplot]

Here, you can only see the blue curve, because the red one is directly underneath it.  Switch the order of the plots (otherplot, oneplot), and reenter to check that they are both really there.  Print out one of these plots (they are all basically the same!) and indicate on it: 1) the points where θ = π/6, π, and 2π; and 2) the direction along the curve in which θ increases.

The command PolarPlot can be used to plot any curve described by a polar equation of the form r = f (θ).

Examples

Enter these commands, and examine the resulting curves.

PolarPlot[1 - Sin[t], {t, 0, 2Pi}, PlotStyle->Red]

This plots r = 1 – sinθ; the curve is called a CARDIOID (heart-shaped thing).

PolarPlot[Sin[t], {t,0,2Pi}, PlotStyle->Blue]

What is this curve called?

PolarPlot[Sqrt[Sin[2 t]], {t, 0, 2Pi}]

This plots a LEMNISCATE (infinity symbol).  Why does Mathematica complain "does not evaluate to a pair of real numbers at t = 1.8326," etc?

Part 2:  Hypothesizing about Roses

Graphs of equations of the form r = sinnθ, where n is an integer, are called "roses," because they have "petals" that resemble a flower's.  The command below defines a function that will plot roses for any value of n.

rose[n_]:=PolarPlot[Sin[n t], {t, 0, 2Pi}]/;IntegerQ[n]

Enter the definition and then use the command below to create a rose with n = 2.  How many petals does it have?

rose[2]

Create plots of roses with other values of n (try 1, 3, 4, and 5, for instance).  It is possible to create roses with only a few petals, or with many, but there are certain numbers of petals it is impossible to have.  Which numbers are those?  (Hint:  Express the number of  petals as a function of n when n is even.  What if n is odd?)

Part 3:  Area in Polar Coordinates

To visualize how one integrates in polar coordinates, we will use a special package your intrepid instructor has developed for this lab.  The command below will load the package.

<<i:/course/root/math162/polarint.m

Let us attempt to integrate in polar coordinates to compute the area of the unit quarter circle.  Of course we already know its area, what is it? Use Mathematica or your calculator to get a numerical estimate of this value.  And what is the equation (in polar coordinates) of the quarter of the unit circle that falls in the first quadrant? (Include the limits on θ.)

We could underestimate the area by inscribing a triangle in the quarter circle, as the following picture shows:

innertriangles[1,{t,0,Pi/2},1]

What is the area of this triangle?

Or we could overestimate the area by circumscribing a triangle:

outertriangles[1,{t,0,Pi/2},1]

What is the area of this triangle?

Of course, using just one triangle is not going to give a very good approximation to the area, but what if we sliced the quarter circle up in to narrow slivers (like pieces of pie) and approximated each one of those with a triangle?  The following commands will draw the pictures for the case when there are five slivers.

inner = innertriangles[1,{t,0,Pi/2},5]

outer = outertriangles[1,{t,0,Pi/2},5]

Do you believe that, as we increase the number of slivers, we will get better approximations to the area?  The command below will show the inscribed and the circumscribed triangles on the same plot; describe how it justifies your conclusion.

Show[outer,inner]

Unfortunately, the areas of these triangles are hard to compute. (If you don't believe me, give it a try.)  If we try to approximate using right triangles, we can compute their areas.  The command below shows the approximation using 5 right triangles.

righttriangles[1,{t,0,Pi/2},5]

For your report, show that if we approximate the area of the quarter circle using n right triangles, and call π/(2n) = Δθ, then the area of each triangle will be tan(Δθ)/2.  Show that, if this method of approximating area works, then the following limit holds:

Finally, prove that this equation is true by evaluating the limit.  (L'Hopital helps here!)