express-the-limit-as-a-definite-integral-on-the-given-interval-displaystyle-lim_n-to-infty-sum_i-1-3

Question

Express the limit as a definite integral on the given interval. 

$ \displaystyle \lim_{n 	o \infty} \sum_{i = 1}^n [5(x_i^*)^3 - 4x_i^*] \, \Delta x $,     [2, 7]

Frank Lin · Accepted Answer

okay. By comparing with remains on form, you always see that this parties are FX. I start part. So that makes three minute over. The submission should be to grow from two to seven five x cubed, minus for X, the ex.

Stephen Hobbs · Answer

when rewriting a limit as an integral. We could always start with the interval that we&#x27;re interested in, and those were going to be the lower and upper limits to the integral. So the integral from 1 to 3 it&#x27;s how that&#x27;s red. And then by the time we get Thio except by stars, we can go ahead and replace all of this each of those exa by stars with exes. So that&#x27;ll be X divided by X squared plus four and then the Delta X There is the same thing or will change to a D X. And so the idea is, this is adding up an infinite number of rectangles whose heights are determined by this whole expression here and who&#x27;s with is a small amount of X or Delta X, and we transform that into this notation as an integral

Yuki Hotta · Answer

All right, let&#x27;s go ahead and do this problem. So the question ISS you&#x27;re given the limit as an approaches to infinity. The some from Aiko&#x27;s toe want to end off XVIII cleared of one plus x of Ike Uche Delta X okay and X is between two and five inclusive. What is the definite integral form off this expression? So the definite integral, the integral from a to B of ffx the the X this is defined to be the limit of a remonstrance. Some limit as angles to infinity off the some off I equals toe Want to end off f off X I using the same notation Delta X, where Delta X is equal to B minus a over and where the X is defined between. We need to be okay. So long story short. What&#x27;s really happening here is that you have a region between a to B, and then you want to calculate the area under the curve using rectangles where this is the height and this is the width. Graphically speaking, this is what&#x27;s going on. So it just doesn&#x27;t example if I have a shape that looks like this. This is a this is B. I split it into a bunch of rectangles where this is f of X, Delta X is each of these width in the height is f off X I where the X I is each of these points. Okay, so we know that the area under the curve right here the approximation gets better and better the more rectangles we have. So if we use in rectangles, this is going to be the sum. But as we use the limit as an approaches to infinity, we&#x27;re gonna have an exact match. So this is the definition off what the definite integral is. Okay, so all we need to do to really answer this question is to see how this is related to that expression right here. It all boils down into figuring out what the limit off the integration is. What effort X is, and de avec dx is usually just Delta X. So we don&#x27;t need to worry about that part. So here the limit and the some we already know that part is just going to be this s portion of the integral. Now here, X, I is my input. We need to match this as my ex, I What is illegal too? Well, it&#x27;s X I times the square root of one plus X I Q. Actually, not that difficult. It&#x27;s exactly what it looks like. This portion becomes your Delta X. So when I multiply f of X, I with Delta X, I get this portion, which is the area under the curve. That&#x27;s a rectangle. Okay, if we add them all up, it will be the definite integral. And we&#x27;re taking the limit as and goes to infinity. So we know that it is going to be the exact answer. Now what is the what are the limits off the integration it&#x27;s given right here? X is supposed to be from 2 to 5, so that is where we get the expression. The integration from 2 to 5 f of x is X square. Root off one plus x you and the exchanges to DX. And this is going to be the answer to this question.

Stephen Hobbs · Answer

writing this limit here. This limited some as a definite and girl. The first thing we could notices. We could rearrange this expression here. So let me just rewrite that as I to the four and keep end to the fourth as well. What? And then you have another factor of one over end because that one of Rennes RDX term and then we could even rewrite the to the fourth to the fourth as all toe the fourth there. And now we&#x27;re really ready to identify what this limit some is equal to as a definite and a girl. So the one over N is equivalent to the width of the trek tangle. So that is a D X turn. The reason why we know the limit is from zero upto one is the definition of Delta X. So Delta X is equal to the minus. I all over. And and in this case, if you just end up with one over end, that means be an A only differ by one. And furthermore, there&#x27;s nothing in front, um, added to it. So a must equal zero, so be must equal one. So that&#x27;s how we know that those or zero in one. And then lastly, we&#x27;re just left with this expression here, Um, which is this multiplied by I so would be won over end times I or I over end, and that is our X value. So that would just be X to the fourth there. And therefore, this is the rewritten form of what we had up top.

Leon Druch · Answer

okay, we have this limit and I&#x27;ll explain what all these two different little symbols and pieces mean, but we want to take this limit and express it as a definite integral. Um so this is a re months um and we&#x27;re taking the limit of these Riemann sums. Uh so we take the limit as N approaches infinity. Um but n is a fixed number and could be seven and could be 80. Eventually. You want to keep finding these sums as an approaches infinity. Uh but for each particular end, whether you know, and there&#x27;s a million and there&#x27;s a million one and it&#x27;s a million to you&#x27;re taking the sum uh from I equals one up to some number. End what end is uh N is the number of partitions that you break up the interval 01 into. For example, if end was just four, you would be breaking up the interval 01 from 0 to 1 into four parts Yes, So let&#x27;s pretend just for a moment and is four. So you would be taking to some uh from I equals 124 of e uh to the excel 1/1 plus X L one times delta X. Uh and then plus E x 0 2/1 plus X +02 delta X. Let me show you what this means. Let&#x27;s pretend N is for If N is for I&#x27;m going to divide the partition from 0 to 1 into four parts four different sub intervals. So this is the first interval corresponding to I equals one, second interval, I equals two, third interval, I would be three and then the fourth interval I would be four, which is what end would be because we broke it up into four pieces. So E except by what is the exit by ex of I is an X. Value in the I. F Interval. So this is the first interval, so that&#x27;s going to be X one. This is the second interval, pick any point in this interval, Call it X two, pick any point in the third interval, call it X three, pick any point in the fourth interval. Call it X four now four X one. Uh If you look at this, this little piece of the sum basically corresponds to your function. So what function can we see in here? E to the X over one plus X. So that&#x27;s our function that we&#x27;re working with. E to the X over one plus X. So this blue curve is the graph of E T D X over one plus X. Now for X one we have the corresponding point on the graph. And this point on the graph is uh the function value. Well if the X value is X one, then the function value is E to the X. Sub 1/1 plus X. Sub one. E to the X sub one. As the value of this function at this point over one plus X. Sub one. And so what we have is we have a little rectangle. Now the width of this rectangle is delta X. And the height of this rectangle is the E to the X sub 1/1 plus X. Someone. So the area of this rectangle is going to be the height. This E TX tough times to with delta X. So this E T D X stuff E T D X. So by over one plus X. L. By is the area is the height of a rectangle. Delta X. Is to within a rectangle. So what happens is when X is one, this becomes the area of the first rectangle. When I is to remember we&#x27;re going up to four because we broke to partition up into four when I is too Okay, here&#x27;s X. Up to, here&#x27;s the function corresponding to X. Up to so when I is to E to the X. Up to over one plus X. Up to is the height of this rectangle, delta X. Is the width of this rectangle. And so height times with gives you the area of the second rectangle. So we&#x27;re going to do this for a total of n equals four times. And then here&#x27;s XO four. Here&#x27;s the point on the craft. So in the end this some from I equals one to end. We just did the example where and was a total of 1234 partitions. This sum is really just a some of the four rectangles and to some of the areas of these rectangles, uh is an approximation for the area underneath the blue curve. Now, as N approaches infinity, all that simply means is you&#x27;re breaking up this interval from 0 to 1 into smaller partitions. More partitions, uh that will be all smaller in size and so you&#x27;re rectangles will fit better underneath the curve and when you take the some of the areas of all the rectangles, you get closer to the area underneath the curve. Now, uh the area underneath the curve, the limit of these Riemann sums as end goes towards infinity is given by the definite integral. The bounds the boundaries for this integral are the X values from 0 to 1. The function that we would be integrating would be f of X E to the X over one plus X and delta X just basically gets replaced with dX. So if we calculate this definite integral, the integral from 0 to 1 of E T D X over one plus X with respect to X. This definite integral will be the limit of these Riemann sums as an approaches infinity. There will also be the area, the exact area underneath that blue functions curve

Katelyn Chen · Answer

you were given the following limit, you were asked to rewrite as integral. So first of all you know not to take into girl from eight a b of effort vax d x his equal to the limit. It&#x27;s an approaches infinity, You&#x27;re rectangular west, Our own it. Where&#x27;s your summation? The path of a plus are reaching their west times. I so basically, we need to take this limit when used to rewrite it in this form. So we have This is equal to the limit and approaches infinity. We can factor out a one over on people to one of I to four over into that four and then based off of this, we can put it. And so we have the limit in the purchase infinity of one of on icicle to one times zero plus one over on times I over four. So we&#x27;re starting to get closer to visit before mine. Over here. Now, based off of this information, we know that our Rick Singler was equal to one over on which is equal to being minus era all over and and so don&#x27;t we know that me is equal to one and we also know that after backs is equal to extra fourth Now, how do we know that&#x27;s well This over here would be like our axe based off of this temple right here. So then they saw for this information. You know that this is our limit over here is equal to integral from 01 Because we have we know that a sickle Zahran BZ also want of other function. That&#x27;s the 4th 3 found the X.

Frank Lin · Answer

we know our interval for this is gonna be from 2 to 7. Therefore, what we know is that are integral is going to be from 2 to 7 and then we know we have five X cubed, minus four ax and then cause it&#x27;s integral, we have to add d x on the end.

Frank Lin · Answer

get to the point where you were rewriting this as an integral. You could always start from the interval that you&#x27;re interested in, and so that is going to be the lower and upper limit. So there&#x27;s going to be the integral from 2 to 7, and from there it&#x27;s pretty straightforward because you&#x27;re basically just going to replace the exit by stars with exes. And so that will be five x to the third minus four X, and then you also replace the Delta X Abi with just a D. X there as well. So, um, this is going to add up up top here. All of the sorry, all of the rectangles on that have a height defined by that expression and a with defined by Delta X I buy. And that&#x27;s the idea down here is, well, it&#x27;s just a cleaner way. Thio express that as an integral

Frank Lin · Answer

All right, let&#x27;s actually move onto the next question. Hopefully, you understood what I explained right here. And you can refer back to this problem any time you want. Okay? Now the question is going to be He&#x27;s black limit as an approach tested Infinity off the song from my goes toe Want end off when you copy it down correctly. Five x i Q t s cubed minus four x. I squared the entire thing times Delta X, where X is from 2 to 7. And it&#x27;s the same kind of question, right? This as the form of the definite integral. So let&#x27;s recap the important parts, the limit and the some. You know, the sigma notation is basically the Greek large s so it makes sense that when you&#x27;re taking the limit, it becomes this fancy looking s okay, the integration is going to be from A to B. We need to identify what those are. We need to figure out what the function is, the height off that point, and we need to figure out what the excess. And of course, we already know that the Delta X is going to be this guy. Okay, so let&#x27;s figure out what? My effort axis. Very simple. It&#x27;s this guy. So we will have the expression five x Q minus four X squared. I already mentioned that the Delta X becomes your DX. I usually don&#x27;t write it, but some people prefer that you use the parentheses so that it looks better where you know exactly where the integration starts at the Trop. It&#x27;s quite clear if you put this thing, but some people actually like it better that way. A and B are simply the starting point and the endpoint from 2 to 7. So here is your answer. So, in summary, the Riemann some can be represented as the integral from 2 to 7 of five. X cubed minus four X squared TX and this is how you answer this question?

Problem

Express the limit as a definite integral on the g…

Problem 19 Easy Difficulty

Express the limit as a definite integral on the given interval.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n [5(x_i^)^3 - 4x_i^] \, \Delta x $, [2, 7]

Answer

$$\int_{2}^{7}\left(5 x^{3}-4 x\right) d x$$

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$$\int_{2}^{7}\left(5 x^{3}-4 x\right) d x$$

Calculus

Catherine R.

Anna Marie V.

Caleb E.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus

Grace H.

Heather Z.

Kayleah T.

Caleb E.

James Stewart

Calculus