express-the-limit-as-a-definite-integral-lim-_n-rightarrow-infty-frac1n-sum_i1n-frac11i-n2

Question

Express the limit as a definite integral.
$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(i / n)^{2}}$$

Fuzail Shakir · Accepted Answer

Okay, so this is the integral from 0 to 2 off one over one plus X squared DX.

Stephen Hobbs · Answer

some as indefinite Any girl. The first part is one over N that&#x27;s going to be basically the width of each of the rectangle, so we could go ahead and convert that to DX. Um, the next thing we might notice is that we&#x27;re only going from zero upto one. And the reason we know that is because it all rests on the definition of Delta X here. So let me explain that a little bit. It&#x27;s b minus a over and and because that&#x27;s all equal to one of rent, then we can basically assume that be is one and a zero. So that&#x27;s how we end up with zero toe one there aan den. The last thing is that this expression times I will give us the of component here, and that just becomes X. So really the integration and is 1/1 plus X squared. And that would be the rewritten form of the limit on the some from above. There

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

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

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.

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.

Problem 48 Easy Difficulty

Express the limit as a definite integral.
$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(i / n)^{2}}$$

Answer

$\int_{0}^{1} \frac{1}{1+x^{2}} d x$

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James Stewart

Biocalculus Calculus for the Life Sciences

Grace H.

Kristen K.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

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$\int_{0}^{1} \frac{1}{1+x^{2}} d x$

Biocalculus Calculus for the Life Sciences

Grace H.

Kristen K.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Kristen K.

Michael J.

Joseph L.

James Stewart

Biocalculus Calculus for the Life Sciences