Okay, We're gonna work through this problem, um, and be able to draw a graph and find upper sums and lower sums. Um, and so what we have here is that we have a power plant, generates electricity by burning off oil. Um, and this, of course, generates pollutants. And we have, um, the each month, the pollutant release rate in tons per day, given for each month. Um, and we know that we're going to assume a 30 day month. Okay, um and so the first thing we want to do is we want to find the low, um, the upper estimate of the total tons and the lower estimate of the total tons just through just through June. So the first thing we do is kind of graph do a graph. And so here we have the month. So January, February, March, April, May, June. So this is January, February, March, April, May and in June. And then here we have Let's see, 0.1, this is our pollutant release rate in tons per day. And then this is, um, 0.2. Um, and then we have 0.3 to a 0.4 and then I've got up here. I'm way up here so I might have to redo my graph. Let's just redo this a little bit here. Let's go ahead for our sake. Since we're going to be adding to it. Let's do this. 0.2 2.42 point six, 4.8. And then we have one up here. Okay, so in January, we know we release point to February is 0.25 March is 0.27 April is 0.34 which I'm probably way over. May's 0.45 in June is 0.52 So somewhere here. So it kind of looks something like this for at least June, and then we're going to be kind of adding to that. Um and so the first thing we want to do is we want to know the upper estimate for the total tons. Okay. And so the width of our rectangle, So we're the first time we're going to do it is the sense of the increasing function. The upper the upper is going to be where you start on the right side and go left for the upper. So that would be the first rectangle and then the second and then down here to April and there May and then February and then down here to January. Okay, So, um, we know that the width of each of the rectangles is 30 days, and we're going to multiply it by each of the heights. So the upper would be June's height, which is 300.52 Then we're going to add it to the next height, which is 0.45 Then add it to 0.34 then added 2.27 and then 2.25 and that would be the last height right here on February's. And so once I do that, I get a total tons a 54.9 tonnes for my upper estimate. Now we're going to do it again for my lower estimate of my total tents, and my with is still going to be 30 days. But now my lower is actually going to be starting off on the left and we're gonna go right, So that would be the first one second, third, fourth and fifth. So now these green dots are actually my height, so I'm gonna start off at 0.2 in 0.25 in 0.27 0.34 and 0.45 I no longer have that 0.52 in there, and that is 45.3 tons. Okay, so now the last thing we want to do is we want to know. And this is where we're gonna have to kind of add in some other months. I'll probably need a new graph is or we can just do it off. We want to find the best estimate. Oops. Or the best case. Let's do it. Do it this way. The best case, um, when we'll have 125 tons released. So the best case, the worst case would be the upper some because, right, that would be the most we would release. The best case would be the lower some. And so we want to know the month that we get 100 and 25. Okay, so we know that in in, um, by June, I have 45.3 tons. Okay, so now what I need to do is just go ahead. If we do the best case, it's just go ahead and keep adding, So this is in June. So I want to keep adding 30 times that x tons per day to this. And so if I have 45.3 tons, um and that would be my lower show. Then that would be, um I would now do 0.52 times 30 and add that to 45 0.3, and that would be 60.9. So now I want to add it. Add point 0.63 So, basically what I want to do this is through June. And so then what I want to do is take that 45.3 and then add it to 30 times 0.52 and then do 30 times 0.63 because this would give me 60.9. This would give me 79.8, and then I want to do it to the next one, which would be in August, which is 0.7. And that would give me 100 100.8. And the end. We would do whips the next one, which would be 0.81 time 30. Uh huh. So this would be plus 30 times 300.81 And that gives me the 1 25. So this would give me 125.1 tons. Mhm. And that would be my lower some through September. So come about September, and this occurs and September. So come about. September is when I would have, um when I would have 125 point tons. Hope this helps.