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The following article, written in 2025, is a submission to the Summer of Math Exposition 4 (SOME4) contest. It assumes that the reader is fairly good at arithmetic, algebra, and coordinate geometry, as well as being comfortable with some of the basic ideas that underlie calculus. In particular, we’re hoping that the reader can keep an open mind about the use and meaning of infinity in mathematics. Nonetheless, the goal of this article is to take concepts that are normally pretty dense and present them in an intuitive, engaging way. With that in mind, don’t worry too much about whether this is above your skill level; if you’re interested in it, just read it and enjoy!
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Table of Contents
Any student of modern calculus can tell you that the task of finding the area under a curve is called integration. But in days of yore, it was called quadrature: the typical trick was to construct a square (in Latin, quadratus) equal to the area you were looking for. Although new methods were eventually developed, the name stuck.
Medieval mathematicians were able to perform a wide variety of quadratures. In particular, they could find the area under any curve obeying the equation $f(x) = x^p$, where $p$ is a whole number $\geq 0$. When such a curve is defined as a function of $x$, the area under the curve turns out to be a function of $x$ as well, **with a surprisingly simple formula:
$$ \int x^p\,dx = \frac{x^{p+1}}{p+1} $$
This is known as Cavalieri’s quadrature formula, after the mathematician Bonaventura Cavalieri, who first derived it. We can use it to show, for example, that the area under the parabola $x^2$ is given by $\frac{x^3}{3}$.
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Incidentally, if you’ve never seen the above notation before, don’t worry about it. For the purposes of this article, all you need to know is that the expression $\int f(x)\,dx$ means “the area under the curve whose shape is defined by$f(x)$.”
Gray text boxes like these contain brief digressions into topics we thought were interesting and tangentially relevant. They can be skimmed, if you want.
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Cavalieri and his contemporaries were even able to go beyond positive integer exponents, finding the area under curves of the form $x^{-p} = \frac{1}{x^p}$ and $x^{p/q} = \sqrt[q]{x^p}$. Amazingly, it turns out that the rule of “add $1$ to the original exponent, then divide by the new exponent” applies to all of these curves too… all, that is, except one. Cavalieri’s formula fails in a single special case:
$$ f(x) = \frac{1}{x} $$
This function of $x$ is known as the reciprocal function. Geometrically, its graph is that of a hyperbola with perpendicular asymptotes.
We’ll spend the rest of this article trying to mathematically describe the area under this curve. At first, the quadrature of the hyperbola might seem pretty niche, more like a historical curiosity than anything. But in the end, we’ll find that it is a spectacularly important problem—one whose solution has implications for the very foundations of calculus and modern mathematics itself!