# 7 – 07 L The Integral Area Under A Curve H1 V2

When you calculate the area under a curve, what you’re actually doing is taking an integral. And the mathematical symbol for the integral is this stretched out “S” looking shape. And often it’ll have what are called bounds of integration at the bottom and the top. So, in this case, t1 is the lower bound and t2 is the upper bound. And the integral is the opposite of the derivative. So, since velocity is the derivative of displacement, this means that displacement is the integral of velocity. Let’s go back to the graph. Now remember that the area under this graph gives us the total displacement. Now I want to show you how to express this effect in integral notation. The integral that expresses this exact same statement looks like this. And the way I would read this is by saying delta x equals the integral from two to four of 100 dt. And that’s a lot. So, let me try to explain what this means. It won’t all be perfectly clear just yet, but that’s okay for now. Total displacement. Well, that we just represent with delta x. And that should be pretty familiar. Let’s take a look on the right-hand side. Well, on the right-hand side we have this integral sign with the bounds of integration. The 100 and the “dt” to the right of it. Let’s talk about this “dt” for a second. There’s always going to be a “d something” in an integral, and whatever that something is, in this case it’s “t”. It’s just meant to tell you what these bounds of integration actually mean. So the fact that the something here is a “t”, well, it means that this lower bound of integration is t equals 2 and not x equals 2 or v equals 2 or anything else. Now all that’s left is this 100, and the reason this just happens to be a simple 100 is because the velocity function was so simple. If on the other hand, the velocity function were more complicated, like let’s say 50 plus 20 times t then, instead of 100, we’d have 50 plus 20 times t. And in general, this v(t) can really be anything. So we can write this generic version of the integral and this would work for any weird velocity function. Now if this all feels pretty overwhelming that’s normal. For now I just wanted to introduce you to a notation that you’re going to be seeing throughout your self-driving car career. And fortunately, computers make using this notation pretty straightforward.