GoKawiilOver the years, mathematicians, logicians and computer scientists have developed various calculi. If you have a background in computer science, you’ve likely heard of the lambda calculus, a model of computation that was developed by Alonzo Church. If databases are more your thing, then you’ve been exposed to the relational calculus without even knowing it, since SQL is based on the relational calculus. If you are into formal methods, then you’ve worked with the predicate calculus, better known as first-order logic. Finally, if you enjoy reading academic papers on programming languages, you’ve almost certainly run into the sequent calculus. However, when someone says “calculus” without modification (e.g., “I’m taking calculus next semester”), there’s no ambiguity about which calculus they are referring to: it’s always one particular calculus. Or, rather, two calculi that happen to be deeply related to each other: differential calculus and integral calculus.
Visually, you can think of differential calculus as being about calculating the slope of a function at a given point. For example, consider this graph:
You might ask, “how quickly is this curve changing when x=6?” In other words, what is the slope of this function right in a neighborhood very close to x=6?
Differential calculus enables you to compute the slope of a function at a given point
Integral calculus, on the other hand, is about the area under the graph over a particular interval. For example, you might ask “what is the area under this curve between x=2 and x=7?
Integral calculus enables you to compute the area under a function over a given interval
If you study calculus, you’ll first be taught differential calculus (sometimes referred to “Calculus 1” or “Cal 1”) and then you’ll be taught integral calculus (“Cal 2”). When you study differential calculus, you learn the rules for calculating the derivative (slope-at-a-point) of a function. And it turns out that it’s quite straightforward to calculate a derivative, no matter what type of function it is. It’s just an algorithm, which means you can easily program a computer to compute derivatives if you wanted to. (As an aside, automatically computing derivatives is a fundamental element in the process of training LLMs. If you’re curious, look up automatic differentiation).
And then, you get to Cal 2, and you learn about how to compute an integral (area-under-a-curve). You will soon discover that, unlike in Cal 1, there is no algorithm for computing the integral of an arbitrary function. Instead, what you learn is a bag of tricks on how to compute integrals for different kinds of functions. You also learn that for some functions, there’s no closed-form solution at all for the integral! As an example, consider the Gaussian function, which shows up in the normal distribution. With zero mean and unit variance, it looks like this:
1 2 π e − x 2 2 \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}
The infamous bell curve
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