Riemann approximation introduction
What we're going to try to do in this video is approximate the area under the curve y is equal to x squared plus 1 between the intervals x equals 1 and x equals 3. And we're going to approximate it by constructing four rectangles under the curve of equal width. So let's first think about what those rectangles look like, so four rectangles of equal width. So it looks like that, like that, and like that. And I haven't really defined the top of the rectangles just yet. But let's think about what those widths have to be if they're going to be equal width. And we can call that width delta x. So this distance right over here, we're going to call that delta x. So delta x is going to have to be the total distance that we're traveling in x. So we finish at 3. We started at 1. And we want four equal-width rectangles. So it's going to be equal to 1/2. So for example, this first interval between the boundary between the first rectangle and the second is going to be 1.5. Then we go 1/2 to 2. Then we go to 2.5. And then we go 1/2 to 3. Now, let's think about how we'll define the height of the rectangles. For the sake of this video-- we'll see in future videos that there's other ways of doing this-- I'm going to use the left boundary of the rectangle to define the height-- or the function, I should say. I'm going to use the function evaluated at the left boundary to define the height. So for example, for the first rectangle, this point right over here is f of 1. And so I will say that that is the height of our first rectangle. Then we go over here to the left boundary of the second rectangle. We're now looking at the function evaluated at 1.5. So that is f of 1.5. That's the height. And so we get our second rectangle. Then, we get-- I can keep going like this-- we get, for this third rectangle, we have the function evaluated at 2. So that's right over here. That's f of 2. And so then we get our third rectangle. And then, finally, we have our fourth rectangle, the function evaluated at 2.5. So the function evaluated at 2.5 is the height. So this is f of 2.5. Remember, in each of these, I'm just looking at the left boundary of the rectangle and evaluating the function there to get the height of the rectangle. Now that I've set it up in this way, what is the total approximate area using the sum of these rectangles? And clearly, this isn't going to be a perfect approximation. I'm giving up on a bunch of area here. Let me see if I can color that in with a color that I have not used. So I'm giving up. I'm giving up this area. I'm giving up this area. I'm giving up that area. I'm giving up that area there. But this is just an approximation, and maybe if I had many more rectangles, it would be a better approximation. So let's figure out what the areas of each of the rectangles are. So the area of this first rectangle is going to be the height, which is f of 1, times the base, which is delta x. The area of the second rectangle is going to be the height, which we already said was f of 1.5, times the base, times delta x. The height of the third rectangle is going to be the function evaluated at its left boundary, so f of 2-- so plus f of 2 times the base, times delta x. And then, finally, the area of the third rectangle, the height is the function evaluated at 2.5, so plus-- that's a different color than what I wanted to use. I wanted to use that orange color-- so plus the function evaluated at 2.5 times the base. This is going to be equal to our approximate area-- let me make it clear-- approximate area under the curve, just the sum of these rectangles. So let's evaluate this. So this is going to be equal to f of-- it's going to be equal to the function evaluated at 1. 1 squared plus 1 is just 2, so it's going to be 2 times 1/2. Plus the function evaluated at 1.25. 1.25 squared is 2.25. And then you add 1 to it, it becomes 3.25. So plus 3.25 times 1/2. And then we have the function evaluated at 2. Well, 2 squared plus 1 is 5, so it's 5 times 1/2. And then finally, you have the function evaluated at 2.5. 2.5 squared is 6.25 plus 1. So that's 7.25 times 1/2. And just to make the math simpler, we can factor out the 1/2. So this is going to be equal to-- write 1/2 in a neutral color-- 1/2 times 2 plus 3.25 plus 5 plus 7.25, which is equal to 1/2 times-- let's see if I can do this in my head. 2 plus 5 is easy. That's 7. 3 plus 7 is 10. And then we have 0.25 plus 0.25, so it's going to be 10.5 plus 7 is 17.5. So 1/2 times 17.5, which is equal to 8.75, Which, once again, gives us an approximation. And clearly, the way I've drawn it right over here, for the function we're using, it's going to be an underestimate, because we've given up all that pink area that I had colored in before. It's an underestimate, but it's an approximation of the area under the curve. In the next few videos, we're going to try to generalize this to situations where we have an arbitrary function and we have an arbitrary number of rectangles. And we'll also start-- in videos after that, we'll look at rectangles where we define the height not by the left boundary, but by the right boundary, or by the midpoint. Or maybe we don't use rectangles at all. Maybe we might use things like trapezoids. Anyway, have fun.