If $ f(x) = x^2 -4 $, $ 0 \le x \le 3 $, find the Riemann sum with $ n = 6 $, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.
using the graph y equals X squared minus four, which is in green. We want to first find what Delta axes, which is the width of each of our Remond rectangles. And we're going to use B minus A, which is three minus zero, all divided by N. We're using sex rectangles here. This gives us 0.5 or one half. So therefore, that tells us that each of our rectangles is going to have a with a 0.5. But what determines the height is actually the midpoint. So this first one, the height is determined by X 0.25 the midway point between zero and 0.5. So if we come straight down, hit the graph, then you just then you just go over both directions, and all of that will be our first rectangle here. The re months, um, is adding the areas under the curve. The next one will be at 0.75 You come straight down, go over and then you connect it back up. That will be our next rectangle and so on. So let me go ahead and give a shortcut here, so we know that we have a width of 0.5. So if we add all of these, we've got that. Then let's go ahead and see what we're plugging in while we're plugging is your 0.25 and we want to find the height, which we would plug into this so we'd square it and then subtract four. I'm going to go ahead and just wait on all the subtracting fours and clump those together. So then let's go ahead and add the next height, which is at this point here, which is 0.75 notice at this point to get from one. So the next you just go over 0.5, by the way, and let's keep going. Plus again, we're gonna wait to clump. All the minus four is at the end there, um, 1.25 where plus 1.75 squared plus 2.25 word and then finally are six height is going to be 2.75 word. So let me just show you that would be right there. Come straight up and then over each direction, back down again. And so that would be basically are six rectangle with you. Go ahead and add on these other ones here. Look like this. Uh, let's see. What would that be? Um, the 2.251 Yeah. And then you have the 1.751 which is here. That would be a negative area under the curve on. Do you have the 1.251? Which would be this. Sorry. It's very hard toe see individually here. But I really hope the idea kind of stick. So there we go. We got four rectangles which are below the X axis. Those will be negative for two to the right. But you're positive for a total of sex like we wanted here. Okay, Now back to the formula that we want to subtract 46 times so you could just do that. And all of this will give us the area. It's the same thing, Is adding length times with for all of these and and adding them up. So this will end up giving us a decimal ended up getting negative 3.6 or as an exact fraction. Negative. 49 over 16. So this represents the re months. Um, which is the area of all six of these rectangles added together, representing an approximation for the area under the curve of X squared minus four from 03