Calculus is Weird. Why Is the Area Related to the Slope? Fundamental Theorem Explained.

Photo: Rhett Allain

In case you didn’t know, calculus is weird. In the intro class, there are really just two big ideas: derivatives and integrals. We can say that the derivative of a function is a way to describe the slope at different points. That’s cool. For integration, it’s essentially the area under the curve of the function. Nice.

Now for the crazy part. The area under the curve is the opposite of the slope. OK, that’s not exactly true — but an integral is an anti-derivative. So if you have a function and take the derivative and then integrate — you get back to where you started. This means that for integration you can just evaluate it by thinking about undoing a derivative. Trust me — this is much easier than actually finding the area as an infinite sum of infinitely tiny boxes.

So, this is the Fundamental theorem of calculus. We could write this as the following:

This might look weird, so let me explain. Suppose I have some function f(). If you integrate this function from some constant (a) to a variable endpoint (x) you will get something that is a function of x (the endpoint). We can call this result g(x). Now, if you were to take the derivative of g(x) with respect to x you would get the function f(x). Or, in other words the derivative of the integral is…