Definite integrals intro | Accumulation and Riemann sums | AP Calculus AB | Khan Academy

What we're going to do in this video is introduce ourselves to the notion of a definite integral. With indefinite integrals and derivatives, this is really one of the pillars of calculus. As we'll see, they are all related, and we'll see that more and more in future videos. We'll also get a better appreciation for even where the notation of a definite integral comes from.

So let me draw some functions here, and we're actually going to start thinking about areas under curves. Let me draw a coordinate axes here; so that's my y-axis, this is my x-axis. Actually, I'm going to do two cases. So this is my y-axis, this is my x-axis, and let's say I have some function here. So this is f of x right over there, and let's say that this is x equals a. Let me draw a line going straight up like that, and let's say that this is x equals b, just like that.

What we want to do is concern ourselves with the area under the graph, under the graph of y is equal to f of x and above the x-axis, and between these two bounds, between x equals a and x equals b. So this area right over here, and you can already get an appreciation. We're not used to finding areas where one of the boundaries, or as we'll see in the future, many of the boundaries could actually be curves. But that's one of the powers of the definite integral and one of the powers of integral calculus.

So the notation for this area right over here would be the definite integral. We're going to have our lower bound at x equals a, so we'll write it there. We'll have our upper bound at x equals b right over there. We're taking the area under the curve of f of x, f of x, and then dx.

Now in the future, we're going to, especially once we start looking at Riemann sums, we'll get a better understanding of where this notation comes from. This actually comes from Leibniz, one of the founders of calculus. This is known as the summa symbol, but for the sake of this video, you just need to know what this represents. This right over here represents the area under f of x between x equals a and x equals b. So this value and this expression should be the same.