Beautiful Minds

This summer, in my preparation for AP Calculus BC, I came across a concept in unit 6 on the area under a curve known as Riemann Sums. The idea that this area could be approximated with two sets of seemingly random rectangles was surprising to me, for I was very familiar with derivatives at this point but had no experience with integrals. However, this foreign concept hasn’t been revolutionary for a long time; indeed, it was invented more than a century and a half ago by one Bernhard Riemann. 

Bernhard Riemann was a German mathematician born in 1826 who accomplished many great feats despite his short life-span of 39 years. He was born in a poor family and had to scrap for every opportunity he could get. Fortunately, his great aptitude in mathematics did not go unnoticed. He graduated from the University of Göttingen with a degree in mathematics and taught for several years. Although he was a genius by every stretch of the imagination, he was ironically known as quite a poor teacher. He bounced from one bad job to another until he contracted tuberculosis and passed away. On the whole, though, he accomplished infinitely more in the field of mathematics and sciences than several of his contemporaries in double the amount of time. 

Riemann is most famous for his conjecture labeled the “Riemann Hypothesis”, which is a postulate that states that all the zeroes of the zeta function (a complex function which is a summation starting with 1 + 2^−s + 3^−s…) lie either on a value where x equals an even negative number or ½ when represented on a Cartesian plane. To this day, the hypothesis has neither been proved or disproved, but its implications are far-reaching, with the validity of many other postulates hinging on the verification of the Riemann hypothesis. 

Nevertheless, I believe that the practicality and applications of Riemann’s direct contributions to Calculus, Riemann Sums and the Riemann integral, are even more pronounced in the present. A Riemann sum is a method used to approximate the area under a curve by dividing the area into a series of rectangles and summing their areas, and they relate directly to definite integrals. 

Integrals are used in many different fields such as in physics, where they are used to measure object displacement and velocity under certain conditions, and economics, where they can find the area under a supply and demand curve. Integration wouldn’t be possible without its parent definition, though, much like the limit definition of a derivative which we take for granted. Integrals are defined by taking the limit of a Riemann sum approximation of a curve and setting it to infinity. In short, integration would not be applicable to the many areas of study as it is today without the advent of Riemann sums. 

The Riemann integral serves a similar purpose, giving us the exact value of the area under a curve. It’s defined as the limit of a Riemann sum as the number of rectangles increases infinitely and their width shrinks to zero, and it formalizes the idea of accumulation: not just with areas, but any quantity that builds up gradually, like distance, mass, or total cost. Thanks to Riemann’s framework, we can define and compute these totals with precision, even for curves that aren’t easy to work with geometrically.

Riemann’s legacy lives on in every application of integration today. The ability to find the area under a curve starts with his simple yet powerful idea, approximating with humble rectangles. Without Riemann sums, integration as we know it wouldn’t exist. His work proves that even the most abstract but simultaneously trivial-looking math can have lasting impact. As Riemann himself once said:

“If only I had the theorems! Then I should find the proofs easily enough.”

— Bernhard Riemann, Proofs and Refutations (Lakatos, 1976)

Works Cited

O’Connor, J. J., and E. F. Robertson. “Bernhard Riemann – Biography.” MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of St Andrews, https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/.

“Bernhard Riemann.” Encyclopaedia Britannica, Encyclopaedia Britannica, Inc., https://www.britannica.com/biography/Bernhard-Riemann.

“Riemann Hypothesis.” Encyclopaedia Britannica, Encyclopaedia Britannica, Inc., https://www.britannica.com/science/Riemann-hypothesis. 

Ruch, D. “The Definite Integrals of Cauchy and Riemann.” Triumphs of Analysis, Ursinus College Digital Commons, 2017, pp. 1–12, https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?article=1011&context=triumphs_analysis. 

Pawlowski, Rodrigo López Pouso. “Riemann Integration via Primitives for a New Proof to the Change of Variable Theorem.” arXiv, 30 May 2011, https://arxiv.org/abs/1105.5938.

“Riemann Integral.” Wikipedia, Wikimedia Foundation, last edited 2025, https://en.wikipedia.org/wiki/Riemann_integral.