About this video
The video introduces the definite integral as the area under the graph of y = f(x), above the x-axis, between x = a and x = b. It explains the notation: the integral sign (Leibniz's summa symbol) with lower bound a, upper bound b, f(x) and dx.
You'll be able to
- describe the region a definite integral represents.
- read and write the notation for the definite integral from a to b of f(x) dx.
- identify the lower and upper bounds of a definite integral.
Key words
- definite integral
- A number that gives the area under the graph of f between x = a and x = b.
- lower bound
- The starting x-value of the integral, written at the bottom of the integral sign.
- upper bound
- The ending x-value of the integral, written at the top of the integral sign.
- summa symbol
- Leibniz's name for the tall S-shaped integral sign, hinting that it means a sum.
Transcript
- [Instructor] What we're gonna do in this video is introduce ourselves to the notion of a definite integral and with indefinite integrals and derivatives this is really one of the pillars of calculus and as we'll see, they're all related and we'll see that more and more in future videos and 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 gonna start thinking about areas under curves. So, let me draw a coordinate axes here, so that's my Y axis, this is my X axis, actually I'm gonna 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 F of X right over there and let's say that this is X equals A and let me draw a line going straight up like that and let's say that this is X equals B, just like that and 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.
And so, the notation for this area right over here would be the definite integral and so, we're gonna 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, this 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
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