I saw polar coordinate calculus for the first time in 1977. I’ve taught calculus as a TA and as a professor since 1987. And yet, I’ve never thought of this simple little fact.
Consider . Now it is well know that the area formula (area enclosed by a polar graph, assuming no “doubling”, self intersections, etc.) is
Now the leaved roses have the following types of graphs: leaves if is odd, and leaves if is even (in the odd case, the graph doubles itself).
So here is the question: how much total area is covered by the graph (all the leaves put together, do NOT count “overlapping”)?
Well, for an integer, the answer is: if is odd, and if is even! That’s it! Want to know why?
Do the integral: if is odd, our total area is . If is even, we have the same integral but the outside coefficient is which is the only difference. Aside from parity, the number of leaves does not matter as to the total area!
Now the fun starts when one considers a fractional multiple of and I might ponder that some.