The Polar Plot Thickens

Some quick examples of Polar Plots in Mathematica. The important thing to bear in mind is that the functions being plotted are r as a function of ; that is, we are plotting r(). Here are a few quick examples.

First up, arcs of circles; in other words, we are holding r constant, in this case, r = 3. (As mentioned on Monday.)

PolarPlot[3,{x, Pi/4, 3Pi/4}]

For this next, I wondered if I could reproduce what happens when we hold  constant and let r vary. It’s a fail, but it’s a fail that was instructive...in the sense that it produced an attractive picture! The best kind of fail.

PolarPlot[x,{x,-10,10}]

This is the integration problem from the book. One part was to plot the domain. I’m sure that this can all be done much simpler in Desmos, but I’m more conversant with Mathematica--not sure how to produce annotated examples in Desmos, for example. (I did play around for a minute or two in Desmos, and it does have ‘polar graph paper’ and I was able to produce the graph below. I’ll show you in class on Friday, but in the meanwhile, if you click the monkeywrench, then select the polar grid, then enter the command:

r = 2 - sin()
(where you type out theta and it becomes )

then below appears this:
__  ≤    ≤ __

and you fill in the domain. For this problem, I typed -pi/2 and pi/2 to fill in the blanks. Definitely easier! Not sure Desmos integrates, but WolframAlpha and other programs will.

Anyways, with that long preamble, here’s Mathematica’s plot.

PolarPlot[2-Sin[x],{x,-Pi/2 , Pi/2}]

And finally, here’s Mathematica’s integration.

Integrate[(2-Sin[x])^2, {x,-Pi/2 , Pi/2}]