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Riemann Sums Hidc Question InformationTextbook [Videos LIApproximate the arca under the curve subdivisions Previcw1" fromIO IuSIng Right Endpoint upproximation...

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Riemann Sums Hidc Question InformationTextbook [Videos LIApproximate the arca under the curve subdivisions Previcw1" fromIO IuSIng Right Endpoint upproximation with 4Get help; Midee]Wlcerm{Points possible: Unllmited attempts _ Message instructor about this questian

Riemann Sums Hidc Question Information Textbook [ Videos LI Approximate the arca under the curve subdivisions Previcw 1" from IO I uSIng Right Endpoint upproximation with 4 Get help; Midee] Wlcerm{ Points possible: Unllmited attempts _ Message instructor about this questian



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Calculate the Riemann sum $\sum_{i=1} ^ {n} f\left(\bar{x}_{i}\right) \Delta x_{i}$ for the given data. $f(x)=4 x^{3}+1 ;[0,3]$ is divided into six equal subinterval $\bar{x}_{i}$ is the right end point.

Right. Let's first try to graph and see what this function looks like. Well, it should look a little something like this whether interception interception is one and that's it for part B. Ah, we need our daughter X. Well, let's first put in our domains that zero and four. And since we want four petitions, let's divide our domain into four. And now we can see that after we put that takes here, our daughter ex turns out to be one. And from here we can we can see what eggs ill is. Eggs. Ill is zero x one is one x two is two x three three and lastly, x four is four park si now what? Whoa! It looked like every word to apply the left and right Riemann some. Well, if we were to apply the left, we want some. It will look like this where we use the left value off each petition and use it to find the height off each column. So this is the left and the right. Whoa, look like this where we use the right off each petition to estimate the area under dysfunction. Now, indeed, in order to calculate the sums that let's do this left, we will need Let's do some equals the X Sorry Dota x times f of zero which is this value over here, plus f off one plus f off to plus F off three. And now if we substitute all of our numbers and so don't her ex is just one f zero is just one, and it's a reminder we have ffx equals X plus one. So f off one is to f off to is three and effort three is four. Now, when we add all of that together, we should get 10 for the right remind some daughter X moves to need Debtor X And now let's go back up. We will use the left. Sorry, we use the right side off each petition. So for the first petition, we use f off one add f off to F of three and F four. Okay, again, Delta X is just one f off. One is to f of two is three after three is four and every four is five. And when you add all of this together, you should get 14 Going back to our graphs appear in C we can see that the re months, um, for the left, some actually under estimates the area under the curve while the right women some over estimates the area under the curve.

In order to evaluate the integral here with a three months. Um Then the first thing we're going to do is find out what Delta X is. B is four a is six negative negative turns that into a positive, all divided by and the number of sub intervals, which is five. So this turns into plus, so that's 10/5 or two. So we're going to use the heights determined by the right endpoints. So the first height that we're gonna actually use and thes we're all gonna have wits of two. So let me just go ahead and put those in here first. We're going to go by twos. In other words, Oh, this is clearly not drawn to scale. Okay, so we're gonna go down from four and then over, and that will be the first area second area determined by negative, too. Third area determined by zero. Then the next area determined by positive, too. And then finally, by positive or here, one way to write that says you could just factor to out. And then we're really adding half of negative four. Pull us, uh, both negative too. Plus F zero all that I'm doing in this formula is adding the with times the height. So the width is too, which is why that's factored out front. And the height of our first direct angle is f of negative four. You multiply those together and it gives you the area. And I'm just adding up all 12345 of these rectangle areas here. So the next one would be plus f of to and then finally plus f of four. Now, what's nice about the function that were given X minus one is we could actually just do this by hand without even using a calculator. Really? So we've got to Okay, half of negative four. Just plug in negative for here. So it's negative for minus one or negative. Five would be this first height here than f of negative two. Is this height for this rectangle here? That would be negative three. And then we have one more negative F zero. We plug in, we get minus one, Then we get f of to this height. I did not eclipse Go back. Not that height. This height here would only just be one. And then finally f of four and that would be plus three. So as you can see these simplify quite nicely. Negative three. Positive three Negative one positive one All cancel two times negative. Five. Therefore, the right Remond sons end up giving us a new estimated area of negative 10 for the area under the curve of why equals X minus one from negative six out to four. So I hope that was helpful there.

To calculate 04 which is the left handsome. We will need Delta X, which is the difference between our access. So that's still 0.5. And I will add up all of the F values except for the last one, because we want the left values of each petition. So this will be sorry. Not zero will be five plus three plus two plus one, and this will give us 5.5 are four will be is still 0.5. Because it's still the same. Dr X now will add all of the F values except for the left, most one on the table. So it will be three plus two plus one plus one, and this should give you 3.5.

Okay, let's start with a where we will draw the graph. If you put this into York, how come later when you're graphing calculator, you should get something like this. This would actually touch the y axis. And here you get one and three for your domain is next. We can we want to part B where we can find our Don't her ex before we. Before we can do that, we should divide our Domi into five parts. Let's do that. 12345 And then we can feel in what the's ticks of. This is 4.1 point four 1.8, 2.2, 2.6. From here we can see what daughter exes that exist this distance here. So that would be 0.4. Now we have x zero, which is one x one is 1.4 x two is 1.8 x three is 2.2 x four is 2.6 and lastly, we have X five is three. Here. We can draw what? I was some slip like the birds singing nicely. Yeah, I only draw the parts where the domain is. So this is three. This is one thing is three again. Before our left some, we will use the right value. The left value. Sorry to calculate the height off each petition, so we will see now. Well, it looks like for our right for all right, some it will look like this where we use the right values of each petition to calculate the height of each column like so as you can see, the left, the left hand some underestimates the the every under the cuff. Where's the right? Handsome over estimates the area under the cub. Now we can actually calculate what the sums are. Let's write again. What? Our function is okay for our left, Some we need our daughter X whoops May by bad. And now we were used f evaluated are using the number on the left of each petition. So for the 1st 1 petition, we have half off one. The next we have f of 1.4 makes is f of 1.8. Next is F off 2.2 and lastly, F 2.6. Now when you substitute these values in our daughter X is zero point for F off one is Ellen and four close. Olin five 0.6 plus Ellen 7.2 plus F off 8.8 and finally have, uh, 10.4. Delta x is still do 0.4. Are you some All of this together in your calculator, you'll get 9.6 and we multiply that together in your calculators. You get 3.84 for our rights. Um, same thing we need our daughter X. And now we will evaluate F on the right off each petition to be our heights. So have l. N of 5.6 plus l and, um, 7.2 plus and and oops, our land of eight point eight. I've just realized this is not actually f. It should be the natural log, Ellen. Okay. Plus, ah, and and 10.4. And lastly. And and 12 If we put all of this together, we should get zero point four times 10.7 zero. They should equal four point 28 And keep this. Keep this in mind. This this over here is actually if of 1.4 and this is F 1.8 and so one


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