Fun with Calculus: Modeling Area Between Curves with Yarn

A major part of calculus includes finding the area under a curve.  Before he or she learns about integration, a calculus student learns about approximating area under curves by means of a method called Riemann sum.  To find more about Riemann sums, visit Khan Academy or Paul's Online Notes.

I noticed, while previously crocheting, that each row resembled a rectangle that could be placed underneath a curve to approximate the area.  If I were to crochet enough rectangles, I would have a Riemann sum approximation of the area under a curve.

I chose two random functions, f(x) and g(x), and drew them on a piece of paper.  As far as I know, these are functions that I just drew on paper and cannot be resembled by any equation.

f(x) and g(x)

I then crocheted a shape to model the area between the curves, which is solved for by subtracting the area of f(x) from g(x).  According to the Riemann sum approximation method, one should choose between a left-hand, right-hand, midpoint, or trapezoidal approximation.  I chose none of those options, since I did not really think about that.  I have a combination of a left-hand and a right-hand approximation.

Model of the area between f(x) and g(x)

Placed on the graph, the crochet model about approximates the area between the curves.  It does not fit perfectly, as it is only an approximation.