Riemann Sum Intro
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Area and Definitive Integrals
Now we're going to start talking conceptually about the approach we're gonna use to find areas and we're gonna use a technique called Riemann Sums. This is named after Bernhard Riemann. Now Bernhard Riemann was a great mathematician. He's actually much more famous for a problem called the Riemann Hypothesis, which currently is the single most famous unsolved question in mathematics.
Now, what this hypothesis says, even just stating the question is well beyond the scope of a first year calculus class. So we're not gonna talk about the Riemann Hypothesis. But I recommend that just if you're interested in higher mathematics. Here, we're gonna talk about Riemann's approach to finding area. And what he suggested was, we can take the area under a function and divide it into rectangles as shown here.
So these rectangles of course, have a thickness of delta x. And then, what we have is a bunch of x coordinates associated with these rectangles. So let's just say for the sake of argument, that we'll take the x coordinate at the left side of each rectangle. So what we get is we have an x1 here, an x2 here, an x3 here dot, dot, dot.
All the way up to an xn. And of course x1 is going to equal just a. X2 is going to equal a plus delta x. X3 is going to equal a plus 2 delta x, and so forth. Then what we're going to do, is take a sum of the area of those rectangles. And so here, the height of the rectangles.
Here what we're doing is we could call this a left hand sum. Now we could do a left hand sum or a right hand sum or a mid point sum, but let's just focus on the left hand sum. What we're gonna say is for every rectangle, we're gonna go to the left side of it, we're gonna use that x coordinate and we're going to plug that x coordinate into the function.
So, for example, that first rectangle, it's going to have a height of x1, f(x1), times delta x. The next rectangle's gonna have an area of f of x2 times delta x, and so forth. Then we're gonna add up all those areas. And so we can write this in summation notation, the summation as i goes from 1 to n of f of x sub I times delta X.
Now, of course, if we use here I've only drawn 7 rectangles. Of course, if we have only 7 rectangles or 10 rectangles, that's going to be kind of a chunky division, and as you can easily see with these rectangles, there's going to be a little area left over. So it's not really gonna do a very good job, is gonna give us a ballpark area. But, the smaller we make delta x and the more rectangles we use, the closer we're going to get to the true area.
And the idea is, if we take a limit as delta x approaches 0 of this sum, that this is going to equal the area. And so this is the basic idea of a Riemann sum. And we will calculate a Riemann sum in the next video.
Read full transcriptNow, what this hypothesis says, even just stating the question is well beyond the scope of a first year calculus class. So we're not gonna talk about the Riemann Hypothesis. But I recommend that just if you're interested in higher mathematics. Here, we're gonna talk about Riemann's approach to finding area. And what he suggested was, we can take the area under a function and divide it into rectangles as shown here.
So these rectangles of course, have a thickness of delta x. And then, what we have is a bunch of x coordinates associated with these rectangles. So let's just say for the sake of argument, that we'll take the x coordinate at the left side of each rectangle. So what we get is we have an x1 here, an x2 here, an x3 here dot, dot, dot.
All the way up to an xn. And of course x1 is going to equal just a. X2 is going to equal a plus delta x. X3 is going to equal a plus 2 delta x, and so forth. Then what we're going to do, is take a sum of the area of those rectangles. And so here, the height of the rectangles.
Here what we're doing is we could call this a left hand sum. Now we could do a left hand sum or a right hand sum or a mid point sum, but let's just focus on the left hand sum. What we're gonna say is for every rectangle, we're gonna go to the left side of it, we're gonna use that x coordinate and we're going to plug that x coordinate into the function.
So, for example, that first rectangle, it's going to have a height of x1, f(x1), times delta x. The next rectangle's gonna have an area of f of x2 times delta x, and so forth. Then we're gonna add up all those areas. And so we can write this in summation notation, the summation as i goes from 1 to n of f of x sub I times delta X.
Now, of course, if we use here I've only drawn 7 rectangles. Of course, if we have only 7 rectangles or 10 rectangles, that's going to be kind of a chunky division, and as you can easily see with these rectangles, there's going to be a little area left over. So it's not really gonna do a very good job, is gonna give us a ballpark area. But, the smaller we make delta x and the more rectangles we use, the closer we're going to get to the true area.
And the idea is, if we take a limit as delta x approaches 0 of this sum, that this is going to equal the area. And so this is the basic idea of a Riemann sum. And we will calculate a Riemann sum in the next video.