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Use the limit of Riemann JUII to determine the area of eich o the following regions. (In each c4S regular partition ad let the smple point $; that the right enelpoi...

Question

Use the limit of Riemann JUII to determine the area of eich o the following regions. (In each c4S regular partition ad let the smple point $; that the right enelpoint of the ih subinterwal )The region unler the curve f(r) = 2'+5 On the interval [-1.4 The region unler the curw f(r) (r -1) on the intenal @5]:

Use the limit of Riemann JUII to determine the area of eich o the following regions. (In each c4S regular partition ad let the smple point $; that the right enelpoint of the ih subinterwal ) The region unler the curve f(r) = 2'+5 On the interval [-1.4 The region unler the curw f(r) (r -1) on the intenal @5]:



Answers

Use Riemann sums and a limit to compute the exact area under the curve. $$y=2 x^{2}+1 \text { on }[1,3]$$

Okay, This question wants us to evaluate the following integral. So to do this geometrically, let's draw a graph. So it tells us that we're going from 0 to 4 and the pieces change at X equals two. So for the first piece, it's just a constant line. Y equals five, but then it turns into a straight line. So again, this is why equals three X minus one. And this is why equals five. So now we see here that we have to regions to find the area of. So first we have this rectangle, and then we have this trap is oId But you should probably find these verdicts ease to help us find the area. So this point, we just see it's on the white y equals five line. So this point is to five. Which means that this point we have to plug in. So why equals three Expose five to find that one. So if y equals three X plus five or sorry, minus one, then why is equal to plugging in 4 12 minus one for 11. So this point is four comma 11. And now we have everything we need to find these areas. So our area, it's just equal to the sum of our two regions cause there's no negative areas. Area one is just a rectangle in area two is a trap is laid, so it's just plug in these quantities. So for the rectangle, we're going to units over and a height of five. And then for the trap is oId are. Height is along the X axis, which moves to units then based one has a height five and base to has a height 11. So our answer is just 10 plus 16 which is 26.

You're gonna find the exact area of this equation using remind some in limits. So first things first, let's find her Delta X and notice that we don't have an end. That's because that's our limit. So we will be solving for the limit as an approaches infinity. So area is equal to the limit as an improve reaches infinity. And we're doing a summation of eyes equal to one when ah, Thanh F uh, and we start off at once a one plus two and I Times two and I will explain these. So this part right here represents how we increase our values, boys. So let's say if we were given a Delta X and it's 0.5 than our answer and go 11.5 to 2.5 to three and notice how it keeps going up. This is our eye right there that represents our I. And so this is just taking the function off those arteries, which we do when we take a re monsoon and we always multiply fire Does X. So I know it may look confusing at first, but it's actually pretty self explanatory at this port. I am just gonna drop the limit and the summation just because it gets tedious. And I'm just gonna solve for this. But no, they're still there. So now we are gonna plug this into our function that we've been given. So four out? Yes, for plus each high. Christine, you're not this you And now I will put the limit back in. And so now this to get rid of the summation, we have a zero. Um, and you should know it. Um, this time we only have I've zero, which is no, I don't have any eyes. And I of one I wouldn't remember apply to usually that's the most have ever seen. So way have to pull a substitute these values for I. So the 1st 1 now becomes 12 and over on the 2nd 1 becomes 16 over and squared times and squared plus and two. And now it's time to just take the limit. Eso This becomes 12 and 16. Inspired over two and squared becomes eat. So our area was people to 20

Okay, So a quick Google search with Dez most Riemann sums will bring you to this website and everybody. If you have an Internet, you have access to this, and all you have to do is type in this function three x squared plus one. Um, and then we're finding the integral our area under the curve between negative one and one. Um and I guess before I get too far ahead of myself, let's get the right number of rectangles and you can count. There's 123456789 10 Rectangles there. Make sure that if you're using this program, you're doing the right Raymond some so writing Sequels. One makes the right. And what you're doing is you. This program is adding up all of these areas together and adding up all these areas together gives us an answer 4.4 Now, if you notice there's some white space in between here and you might be thinking will is this an exact answer? Because this wide space mike, it made up for these over approximations to the right. Well, let's see if we increase the number of rectangles to 20. Oh, we definitely have less white space so less of an under approximation on this one. But we have, you know, still in over approximation on this one. So now, for n equals 20 we get an area of about 4.1 Um, hopes they'd even asked for that. They asked for 30 next 30 rectangles. Okay, Still not exactly four, but 4.4 Um, Then the next one, they asked for 60. So yeah, I'm zoomed in enough. I think that you can see that there's still some under approximation to the left. And then there's some over approximation. But now we're getting about 4.1 So let's change that The 80. Okay, 4.6 And what's nice about this computer program? Because its internet based eso weaken, stop and weaken. Say, Okay, I'm gonna safely assume that this area is so small. These differences air so small this area is probably exactly four, which looks about right like this is an area of one that's the one by one square. Here is an area of one. This area might be exactly one like this area could fill in there. This area could fill in here. But this program, you could actually go upto 800 rectangles, and you don't even see any white space. I really have to zoom in to see that there is an over approximation and in under approximation. Um, eso this program does this really quickly. I could do 8000, and you could see that that answer is just getting closer and closer to zero. Um, so there's a great visual. I go back to the last answer, uh, make that conjecture that the area under the curve will be exactly for

Okay, So in this question, we are given a function over a specific interval. Eight were adds to calculate the right Riemann. Some given four different numbers of seven rolls. So I've got the answers, written it and table over here on the left. Ah, let question asked for And, um what weekend? How we need to go get a beach value as we first start off by getting Delta X, uh, which is our range divided by our number of seven rolls. And then we need funding, um, our endpoint values for each sub in Rome my okay, are XK values. So X cables, a beginning of the first sub roll, plus k times delta X, the length of each step in the role and then lastly, way to plug in. So we're gonna get expression for that rather than an exact value. We're going to keep ks a, um, variable there, and then we're gonna plug that expression into the function that we were given. Um f so that's FM XK. Ah, and we're gonna sum that up from one to end on end while also multiplying by Delta X. So you do that and you're gonna need your calculator to do that. Um, then you will be able to get the answers on left here in the table and impact be asked us a boat. Um, the approach, the value Reitman some as, and the number of seven year olds purchase infinity. Well, that's even see, um, it's it's not changing. Our values were all 3.14159 approximately. And it actually keeps going because as I'm sure, you guys all realized the opening digits of pi. So, um, as an approaches infinity, um, the answer won't change. The area under curve will continue to be, uh, by


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