 Welcome back everyone. I wanted to talk some more about calculating area under the curve. I actually want to continue the example we finished in the the last video, right? So if you haven't seen that yet, go go back and look at it. What we were trying to do is we were trying to find the area under the curve f of x equals x cubed minus six x as we range from a equals zero to b equals three. We were using six rectangles and we were trying to determine the height of the rectangle using the right points. So what you see in the screen right here is actually the graph of the function y equals x cube minus six x. And you can see here in green and red the associated rectangles to that area under the curve. And so like we saw last time, some of the areas, some of the rectangles lived above the x-axis. Those are going to have positive areas because the y-coordination is positive there. And those are illustrated on this Desmos website in green. And then there are some rectangles have negative area that will be found on those are below the x-axis. And on this Desmos app, they are illustrated here in red. And if you want to play around with this calculator yourself, there's a link below in this video. So feel free to use that. And so this this is actually a nice calculator that can help us do calculate these areas of the curves because as you saw in the last video, we had six rectangles and there was a lot of a tedious arithmetic we had to go through. I didn't even show you all of them because we had to calculate f of 0.5, f of 1, f of 1.5, f of 2, etc. And there was some arithmetic going on there that I had done prior to the video so you didn't even see it. These things can get very tedious very quick and man was not meant to do these type of calculations. That's why man invented calculator. So I wanted to show you some online calculators that you can use to help you out as you try to approximate these areas under the curve here. And so this Desmos one I show you first because I like it because it actually can show you not just the answer but it shows you geometrically what's going on right here. So I'd notice this thing is manipulative, right? So you can type in the function so you have you can type this then you can change that if you want to. You can change the left end point and that affects things. I'm going to put it back to zero right. We'll just leave it there. You can switch the right end point as well and it changes in real time. Of course, you can also change the number of rectangles here. Let's say we want to go up to doesn't want me to change this. Well, I'll just kind of see what it looks like there. You can change the number of rectangles. Something like that. You can also change this value right here. This C value changes the sampling points the Xi star and there's some instructions right here what's going on here. Negative three doesn't even make any sense for rectangles. What are you doing computer? This needs to go from zero to 100 yet. So that's a good number of rectangles. We'll do five at the moment. So going back to the C here just understand what's going on here. C is represented the method of choice. How are you going to choose the Xi star? If you choose C to equal one it computes using the right end point rule. If you choose C to be zero try that again. If you choose C to be zero it'll choose the left end points and if you choose C to be one half it chooses the midpoints of the interval and so this number right here this I right here whichever you're looking at that'll tell you the area under the curve its estimate inside number changes. This is the answer you're looking for. If you're looking for the right end point rule the left end point or the midpoint rule or the left end point rule like this and then I also added the trapezoid rule here where it averages together the left and right end point rules. We talked about each of these in lecture 43. So if we put this back to six and we put this to three and we put this to zero and right end point rule come on you can do it. Oh this web app doesn't really like to work with my recorder very well. It's come alive again. So this right here was the exercise we had done in the previous video and you'll see that the answer we got was the same negative 3.9 375. So this Desmos website does pretty nice for computing these the the so-called Riemann sums. We'll talk about those more in a future video here. It helps this character this area to the curve. Another one I'm going to advertise to you is through emathhelp.net. It has a pretty nice calculator as well. They have a lot of calculators. I like the calculator they have here a lot although my one real big concern with emath help is their ads are extremely annoying and in your face all the time. Fortunately my recording app is kind of having some issues with some of them which is good news for us. When I personally go to this website I have a pretty nice ad blocker maybe you know I know ads are how they make their money but at the same time I don't want what they're selling so feel free to block them to their doom if you have to. But what you can see here it's it's similar to what you saw in Desmos you can enter your function x cubed minus 6x set your a value set your b value set the number of rectangles and you can choose how you're going to choose the sample points and so then there's going to be a step click button down here that says calculate and so it takes a second to calculate it'll reload the page and now you get some new advertisements seriously go to the you no no go to suu come on what are you thinking there internet why does it think I want to go to the you right now anyways and you might not be able to see this super well on this video but if you if you go to the website yourself the link is below you can this will look at a lot this will look a lot better to your screen you can manipulate the sizes and things so it'll repeat itself and tell you that you're trying to find the area under the curve x cubed minus six as you go from zero to three this integral notation will make more sense in a future lecture so just just postpone a little bit there and so you have your function you go from a equals zero to b equals three six subdivisions it then calculates the delta x which you got one half and then calculates each of the x i's so we have the interval zero to one half one half to one one half to three halves three halves to two two two to five halves and five halves to three just like we did before it calculates each of those values f of x one which was negative two point eight seven five f of x two which was negative five you go through all of those so it gives you all of those details and then in the end it'll give you the value one half times the sum of all those things you get the negative three the negative three point nine three seven five so if you're only interested in the final answer you'll calculate that for you but also it shows you each and every step along the way so you can kind of understand where the calculation is coming from if you don't want to see the steps you can actually click where it says show steps and if you turn that off it'll do the calculation again with just the just the final answer there so you know hidden amongst many ads here but negative three point nine three seven five here so what i want to do is just do another example of this uh so let's time let's take the function x squared and we're going to go on the interval zero to four let us do four subdivisions why is this thing stubborn there we go we're going to do four subdivisions so four rectangles and this time i want to use midpoints so for the decimals want to have to choose this c to be point five and so you can see right here if you go from zero i'm sorry not zero to four i want to go from zero to one that's the example i prepared so i'm going to zoom in a little bit so you have your standard parabola y equals x squared and so if you look at the area under the curve using these rectangles you get the four rectangles you see right there their height is determined by the midpoint so we're getting these values right here and then if you scroll down you can see the estimate uh the area is going to be approximately point three two eight one two five that's approximately how good the area is under the curve and like we saw on previous videos the midpoint rule does pretty good because although the midpoint rule at some points might overestimate at other points it underestimates and it kind of averages out to be a pretty good estimate um if we were to do that with the emath emath help calculator we type it in x squared use a carrot there for exponents we want to go from zero to one four rectangles and then we can choose our scheme here there's the left rule the right rule the midpoint rule the trapezoid rule so we'll pick the midpoint rule and i do want to show the the steps here so let's calculate so this gives us all the explanation of what was going on in that calculation right so we're trying to approximate the area of under the curve zero to one x squared we had four rectangles we're using the the midpoint rule here so we're going to calculate each of these midpoints x zero plus x one over two x one plus x two over two x three plus x x two plus x three over two keep on going in this situation your delta x will be one minus zero over four which is one fourth um you're you're getting a four sub intervals and it'll look like zero to a fourth a fourth to one half a half to three fourths and three fourths to one then if you find the midpoint of each of these intervals you're going to get one eighth three eighth five eighths and seven eighths calculate the function at those values since we're just squaring function we have one sixty fourth nine sixty fourth twenty five sixty fourths and forty nine sixty fourths there's the decimal expansions there as well so we have to add together those four values times it by delta x which is one fourth and then you get the the value point three two eight one two five so that's pretty cool uh because these calculators can give you all the details you need um as you're working through these things and again i do want i want people to use calculators these type of numerical approximations should be done with a calculator it's very important that everyone understands exactly what the calculation is what's going on we should know what's going under the hood so to speak but please use technology to help you so you don't spend years upon years trying to calculate these things and i'll talk some more about those in the next video uh look at the link below to find it and please subscribe bye