The Fundamental Theorem of Calculus

In this post, I want to talk about a way to think about the Fundamental Theorem of Calculus that will hopefully give more intuition for it. As a reminder, the Fundamental Theorem of Calculus describes the connection between differentiation and integration, which are the two main operations in calculus. This theorem comes in two parts: one that says what happens when you differentiate an integral, and one that says what happens when you integrate a derivative. Let’s state the two parts now.

First Fundamental Theorem of Calculus: \frac{d}{dx}\int_{a}^{x} f(t) dt = f(x)

Second Fundamental Theorem of Calculus: \int_{a}^{b} f'(t) dt = f(b)-f(a).

Let’s start by talking about the First Fundamental Theorem of Calculus. In this, we are differentiating an integral. To understand what this is saying, let’s break it down into parts. The integral is calculating how much of some quantity has accumulated by a certain point in time (this could be any quantity, like distance traveled, water in a container,…). Then, the derivative is asking about how much the total quantity is changing at a particular moment. For example, how much is the amount of water in the container increasing right this second?

The First Fundamental Theorem of Calculus says that the answer to that question is just equal the function f evaluated at the upper bound of the integral, or f(x).

Let’s think about why this makes sense. The integral is keeping track of how much total has accumulated until a certain point. Then, the derivative comes along and says “how much are you changing right this second?” The answer to that question should just be the current value of the function f at that the time t=x, or f(x). You would at least expect the rate at which the area under a function increases to be proportional to the height of the function, and the First Fundamental Theorem of Calculus just says that the proportionality constant is equal to 1.

The Second Fundamental Theorem of Calculus says that if you want to find the total effect that a function’s derivative has over a region [a,b], you just have to compute the difference in the function’s values at those two points. This makes sense because if someone asks what the total effect of the slope of a mountain was in a region, you would just tell them how high you climbed in that region, or the difference between the height of where you ended and where you started.

Alright, now let’s talk about the fun stuff. Both parts of the Fundamental Theorem of Calculus give connections between differentiation and integration. Even though it’s not too hard to understand some intuition behind why each part of the theorem is true, there’s actually something pretty cool going on here! Differentiation is really asking a local question about a function. It’s asking, “How steep is the function at this particular point?” Whereas integration is really asking a global question about a function. It’s asking “In this region, how much have we accumulated by a certain point?”

It’s interesting that these two topics are related because it means that the Fundamental Theorem of Calculus acts as a way to connect the local behavior of a function to the global behavior of that function!

The concept of local/global statements about an object comes up a lot in Differential Geometry, which studies curves and surfaces. A lot of theorems in Differential Geometry talk about which properties of a surface you can measure from the curve/surface (local) (these are called intrinsic properties) and from observing the curve/surface from a distance (global) (these are called extrinsic properties).

So, the Fundamental Theorem of Calculus is kind of a little peek into how local and global properties of functions are related to each other!


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