 Hello and welcome to a screencast about computing left, right, and midpoint estimates. Today we're going to be looking at estimating the definite integral from 0 to 2 of sine of x squared dx with l4, r4, and m4. So first of all you notice the word estimate. It does not say compute. And that is because we cannot compute this integral using the fundamental theorem of calculus. We cannot do an antiderivative. This problem is kind of set up for a u substitution because I see an inside function here with this x squared. But remember with u substitution you also need the matching derivative in the integral as well, or at least some constant variation thereof. Well the derivative of x squared is 2x. I don't see any other x's in this integral. dx doesn't count because that's what our differential is. So that just doesn't work. So that's why with this one we have to estimate it. Because you can actually put this integral into your calculator or a computer operating system or something and it'll spit out an answer. Where does an answer come from? So that's what we're going to be looking at today on a much smaller scale of course. We're only going to be doing four subintervals, so that's what this little subscript 4 means. But trust me, your calculator is way more than that. But anyway, so what I've done then is I've taken the endpoints on my integral, so that was 0 to 2, and I've cut it into four subintervals. So here we've got 0 to 0.5, 0.5 to 1, 1 to 1.5, and 1.5 to 2. Okay, so I'm going to go and make some left rectangles, and I always like to draw these out first. So it hits my function at 0 right here at 0. It hits my function at 0.5 up here. It's my function at 1 about right here. It hits my function at 1.5 about right here. Okay, and then down here it's also going to hit my function at 2, because we'll need that one later. Oopsie, go away. Alright, so I draw in my rectangles, so this one's going to go over here. So that's going to be nothing. Then I'm going to pop up here. That's going to be my second rectangle. I'm going to pop up here. That's going to be my third rectangle. And then I'm going to pop over here. That's going to be my fourth rectangle. Now I could just eyeball these, but we actually have a function. So we can use our endpoints here, plug them into this function, and actually get out of value. Okay, so L4. I need to do my delta x. Let me put that up here. So remember that's b minus a over n, which in our case is 2 minus 0 over 4, so that's going to give us a half. And delta x is going to be the same for every single one that we do here, because we're not going to be doing any different intervals or notations or anything like that. Okay, so I'm going to factor that one half out. And then now I need to start figuring out where am I evaluating my function at. Well, I'm going to be evaluating it at 0. I'm going to be evaluating it at 0.5 or 1.5. I'm going to be evaluating it at 1. And then I'm going to be evaluating at 1.5, which is the same as 3 halves. Okay, so that means I'm going to be taking these x values and I'm going to be plugging them into my function. So that gives me 1.5. Then I'm going to have the sine of 0 squared, which is 0, plus the sine of half squared, or 1 fourth, plus the sine of 1 squared, which is 1, plus the sine of 3 half squared, which is 9 fourths. Okay, chug all of this into your calculator or whatever other computing systems you've got. And then we come up with a number of 0.933474. Okay, looking at this I would definitely say, I think that's probably an overestimate, if I had to guess. Because this rectangle especially goes way over and then we're missing out on the negative part. So yeah, this isn't a very good estimate, but that's okay. It's about as good as we're going to get for right now. Okay, let me switch colors on us. Let's go to R4. So we're still going to be using the same points, but now we're going to be going from the right-hand side instead of the left-hand side. So this time I'm going to start at 0.5. I'm going to pop up and hit my function. So that's going to come over this way. Trying to make these darker so you guys can see them. I'm going to pop over here to 1 and go over. So there's my rectangle here. I'm going to go to 1.5. So that's like here. Interesting. So that one's almost the same as this one. Almost. And then I'm going to pop down here to 2. So that's going to look like that. Okay. Fantastic. So now let me write out what these are going to look like then. So we've got R4. That's going to be the same 1.5 because that's my delta x times. Now what function values am I going to want to use this time? I'm going to want to use f of a half. I'm going to want to use f of 1. I'm going to want to use f of 3 halves. And then this time I'm going to want to use f of 2. Okay. Because again I'm going from the right. So you can always start over here at the right-hand part of your integral or your interval and then work your way backwards. Okay. Do just like we did up above. And when you crunch this in your calculator then you should end up getting an approximation of 0.555073. Okay. So I got that number by taking these four x values and plugging them into my function to figure out then what the y values are. Okay. Now we've got a big mess here. So let me do some erasing. Oh and the right. Well that one obviously was under. Sorry. Got rid of those. That would be an underestimate just because of how the function worked out. Okay. Now midpoints. So I've still got my 0 to 0.5, 0.5 to 1, et cetera, et cetera. But now I'm going to want to use the midpoint of this interval. So that's going to be like about right in there. Then my midpoint here is going to be about right here. Okay. You can make tick marks on your axis if you want to. My midpoint here is about at my height of my function there. And then my midpoint here is about right in here. Okay. Let me go ahead and draw in my rectangles now. So I go up and hit about right here. This is why we don't use just a drawing because my drawings are not very good. There's my second one. There's my third one. And there's my fourth one. Okay. So I think this one's definitely going to be the best approximation because these go a little bit over, a little bit under, but they kind of, you know, I guess, make it even. So let's write out what this is going to look like. So M4 is a half, again, because even though these rectangles look different than the ones before, they're with the still a half. Okay. Now what about our heights? Well, they're still going to be our function values, but I'm not going to be using 0.5, 1 anymore because I use the midpoints of these intervals. So the midpoint between 0 and 1.5 is 1 fourth. The midpoint between 1.5 and 1 is 3 fourths. The midpoint between 1 and 1.5, that's 5 fourths. And hopefully by now you see the pattern then these are odd fourths. So my next one's going to be 7 fourths. And that makes sense because 7 fourths is halfway between 3 halves and 2. Okay, again, plug these values into your function, in here for your x, and then when you crank that value out, that gives you 0.837369, which definitely by far is our best estimation. If you were to plug this integral into a computer system, operating system, or, you know, anything like that, I bet this would be the best one. Okay. How do I know that? Well, I've been doing this a while. But yes, you can tell just by the picture it's going to be good. Also, another interesting thing to point out is a lot of students typically think that the midpoint is the average of the left and the right. But that's definitely not the case here, right? If you were to average 0.55 and 0.93, I bet that's closer to like 0.7. Okay, this is actually 0.8. So don't just rely on the left end point and the right end point to find your midpoint. You actually want to go through and do the computations by hand. All right. Good luck with these. Thanks for watching.