Suppose that a volcano is erupting and readings of the rate $ r(t) $ at which solid materials are spewed into the atmosphere are given in the table. The time $ t $ is measured in seconds and the units for $ r(t) $ are tonnes (metric tons) per second.
(a) Give upper and lower estimates for the total quantity $ Q(6) $ of erupted materials alter six seconds.
(b) Use the Midpoint Rule to estimate $ Q(6) $.
Upper estimate $=220$ tonnes
Lower estimate $=172$ tonnes
So in this example you're given tea time and r of T. The rate at which solid materials were spewed into the atmosphere. So I have the graph here for you. And then the table. And then I went ahead and plotted the point because I think it's always a good idea to draw yourself a picture so you can see what's happening. So it looks like it's a fairly steady increase. Now you're supposed to find the lower and the upper estimates. So I'm gonna start with lower and then we'll do upper. So when doing this, remember we're thinking about rectangles. So let me get my little graphic here. So each of these rectangles are going to be one unit wide. So our width or a change in T. Is one for both the lower and the output because all of these are one. So that's going to be nice. But the height of the rectangle is going to change depending on which side we're going to be dealing with. So for this one if I start here and use that, why value is the height? There's the rectangle. And then I'm going to use the Y. Value at one for my height And the Y. value at two for my height. It's a little off on my craft but you get the idea and then I would do the same thing for five and six but I'm not going to use six because you can see my rectangle. I've stopped, I've gotten all the area. So this is actually going to give me a lower estimate because I'm not using this area right here. So to do that again, remember the area of a rectangle is length times width or base times height. So if we call this the base then our height would be based on the is found by the Y value. So the area of my lower each one will be one unit wide now because each of them have the same with or the same base. I'm not going to do one times everything. I'm just going to put a one out in the front. And now the Y values are my heights. So the first rectangle was two units high. The Second One was 10, 20 34 44 and 52 Because that's my y value at 5:52. So adding all those up, That gives me a lower estimate of 160 two. So 162 metric tons. Now my upper I'm gonna pause for a second, clear my screen. So my upper estimate is going to give me a value that's a little higher. And when I do this one I'm actually going to start with six And that's going to be the height of my rectangle and then the rest of them will actually be the same except I'm going to be going the other direction. So the Y values at each of these points give us the heights of our rectangles. I know my pictures a little off but you can see now in this situation, I'm not going to use the Y value that went with zero. So I'm not using this point. So all of these again still have a width of one or a base of one. So I'm going to do the one and now I'm going to use all the values except The two. So this height was 10, The next rangel rectangle, the height is 20 and I add all those up. So these values are the Y values which give me the heights of the rectangles. And when you add all those up, You end up with 220 metric tons. Now the next part you're asked to use the midpoint. So I'm going to pause here and clean my screen off. All right, so my the screen is clean now to use the midpoint, I need to find the midpoint between each of these values. Now the midpoint for the tea is pretty easy because halfway between zero and one would be a half half boy, between one and two would be 1.5, 2.5 3.5 4.5 and 5.5. So I'm looking for rectangles now, that will be here using the mid points. So we're actually not going to use those. So each of these are still going to be one unit wide because a half To 1.5 is still one unit wide. So our base is still going to be One second. But now I have to figure out what is this height, what is the Y value here. So now I need to plug it in so I'm going to need to evaluate our at .5 Are at 1.5 Are at 2.5 etc etc. But since I don't have the actual graph, What I'm gonna do is I'm going to find the endpoint or sorry, the actual function and find the midpoint between the Y values. So this would be to plus 10 divided by two. So that's 12 divided by two. And that gives me six. And then I'm going to do 20 plus 10 divided by two And then 34 plus 20 divided by two. And then I'm going to keep going. So are at 3.5 Art 4.5 And are a 5.5. Again I'm finding the midpoint by just adding the two Y values together and divided by two. So I went ahead and pause the video again so that you didn't have to watch all of that. So again with this particular one I'm using the midpoint to determine my height. So my first Value would be about six. So maybe about there. So if I was drawing the representative rectangle that's how high it would be that's a little too wide actually there and then I would make my rectangles again but going from midpoint to midpoint. So now to estimate the area Again, the wits are all one, and now I just need to add these Y values together, so six plus 15 plus 27. And when those are all added together, I have 198. So based on the midpoint, That would be 198 metric tons.