Worked example: finding a Riemann sum using a table

- [Tutor] Imagine we're asked to approximate the area between the x-axis and the graph of f from x equals one to x equals 10 using a right Riemann sum with three equal subdivisions. To do that, we are given a table of values for f, so I encourage you to pause the video and see if you can come up with an approximation for the area between the x axis and the graph from x equals one to x equals 10 using a right Riemann sum with three equal subdivisions, so I'm assuming you've had a go at it, so now let's try to do that together and this is interesting, because we don't have a graph of the entire function, but we just have the value of the function at certain points, but as we'll see, this is all we need in order to get an approximation for the area, we don't know how close it is to the actual area with just these points, but it'll give us at least a right Riemann sum for the approximation, or an approximation using a right Riemann sum. So let me just draw some axes here, 'cause whenever I do Riemann sums, you can do 'em without graphs, but it helps to think about what's going on if you can visualize it graphically, so let's see, we are going from x equals one to x equals 10, so this is one, two, three, four, five, six, seven, eight, nine, 10 and so they give us the value of f of x when x equals one, when x equals two, three, four, when x equals seven, five, six, seven, eight, nine, 10 and x equals 10 and they tell us that when x is one, we're at six, then we go to eight, to three, five, so let me mark these off, so we're gonna go up to eight, so one, two, three, four, five, six, seven, eight and so what we know, when x is equal to one, f of one is six, so this is one, two, three, four, five, six, seven, eight, so this point right over here is f of one, this is the point one comma six and then we have the point four comma eight, four comma eight, we'll put this right about there and then we have seven comma three is on our graph, y equals x f of x, so seven comma three would put us right over there and then we have 10 comma five, so 10 comma five, put this right over there. That's all we know about the function, we don't know exactly what it looks like, our function might look like this, it might do something like this, whoops, I drew a part that didn't look like a function, it might do something like this and oscillate really quickly, it might do, it might be nice and smooth and just kind of go and do something just like that, kind of a connect the dots, we don't know, but we can still do the approximation using a right Riemann sum with three equal subdivisions, how do we do that? Well, we're thinking about the area from x equals one to x equals 10, so let me make those boundaries clear, so this is from x equals one to x equals 10 and what we wanna do is have three equal subdivisions and there's three very natural subdivisions here, if we make each of our subdivisions three-wide, so this could be a subdivision and then this is another subdivision and when you do Riemann sums, you don't have to have three equal subdivisions, although that's what you'll often see, so we've just divided going from one to 10 to three equal sections, that are three-wide, so that's three, this is three and this is three and so the question is how do we define the height of these subdivisions, which are going to end up being rectangles and that's where the right Riemann sum applies, if we were doing a left Riemann sum, we would use the left boundary of each of the subdivisions and the value of the function there to define the height of the rectangle, so this would be doing a left Riemann sum, but we're doing a right Riemann sum, so we use the right boundary of each of these subdivisions to define the height, so our right boundary is when x equals four for this first section, what is f of four, it's eight, so we're gonna use that as the height of our, of this first rectangle, that's approximating the area for this part of the curve. Similarly, for this second one, since we're using a right Riemann sum, we use the value of the function at the right boundary, the right boundary is seven, so the value of the function is three, so this would be our second rectangle, our second division, I guess, used to approximate the area and then last but not least, we would use the right boundary of this third subdivision when x equals 10, f of 10 is five, so just like that and so then our right Riemann approximation, using our right Riemann sum with three equal subdivisions to approximate the area, we would just add the area of these rectangles, so this first rectangle, let's see, it is three wide and how high is it? Well, the height here is f of four, which is eight, so this is going to be 24 square units, whatever the units happen to be, this is going to be three times the height, here is three, f of seven is three, so that is nine square units and then here, this is three, the width is three times the height f of 10 is five, so three times five, which gets us 15 and so our approximation of the area would be summing these three values up, so this would be 24 plus, let's see, nine plus 15 would give us another 24, so it's 24 plus 24, it gets us to 48, there you go, we just using that table of values, we've been able to find an approximation. Now once again, we don't know how good our approximation is, it depends on what the function is doing, there is a world where it could be very good approximation, maybe the function does something like this, maybe the function just happens, let me make it a little bit, maybe the function does something like this, where in this case, what we just did would be a very good approximation or maybe the function does something like this, where in this situation, this might be a very bad approximation, but we can at least do the approximation using a right Riemann sum just using this table.