Math CentralQuandaries & Queries


Question from Dustan, a student:

Awesome site!

Anyways heres my question, I am working on a way to compute very accurate areas for irregualr surfaces by using the idea of a largest possible cicrle, I go in "generations" with gen1 being the circle of mamximun radius enclosed by the bounds of the shape, gen2 is then all circles fomr r-->3r/4.  I continue this and graph the results.

As gen goes higher and higher, the radius of the circles formed becomes less and less but the number of circles formed continues to climb with each successive generation (usually it takes up to gen 3 or 4 to see a deffinate exponextial increase). After I graph the results, with r on the y axis and number of circles created on the x axis my plan is then to integrate the curve.  I want to integrate as far as possible to get as high of accuracy as possible, and thats where the problem arrises, when r get small # of circles gets huge, but r can decrease indeffinatly right?

the circles get smaller and smaller and climb in numbers, how do I give a definate value to the integral of an infinatly decreasing curve?  I was thinking about representing the largest possible circle as a seires that converges to 0 but what is the formuls for a circle of largest possible radius within a closed irregular shape?

Im sorry if this is hard to follow, its finals week here in Idaho, so im kind of frazzled, please let me know what you think.

You can do the same trick with any shape whose area is easy to compute, so why not use rectangles?  The problem with circles is that they do not do a good job of approximating any shape, even another circle!

Suggestion: try your approach on a region whose area you know exactly, and compare the answer you get with your approach using different shapes (circles, rectangles, triangles, trapezoids) to see what shape
gives the best answer with the least work.


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