Question

Evaluate. Riemman Sums

Answer 1

the answer is 2ln2-3/4

Answer 2

i was just wondering how this form got into this form?\[\frac{ 1 }{ n^2 }\sum_{i=1}^{n}(n+i)(\ln(n+i)-\ln n)=\frac{ 1 }{ n }\sum_{i=1}^{n}\frac{ n+i }{ n }\ln \frac{ n+i }{ n }\]

Answer 3

Properties of logs (quotient rule) makes ln a - ln b = ln(a/b)
and they can move n's in and out of the sigma, but not i's ... so they brought one of the n's in from the denominator in front.

Answer 4

I suspect that this is a telescoping series... but I've never seen anything quite like it.

Answer 5

oh thanks i can solve it now lol

Answer 6

i think we should make it into integration?

Answer 7

(correction: i think i can solve it, not i can solve it)

Answer 8

Make it into integration? I don't think that's allowed here, unless you're not telling us the whole question. Since this is riemann sums I imagine that you're really doing this to somehow approximate an integral, but I don't really know what that is.

Answer 9

One rule that may help is that "if the sequence of partial sums converges, the series converges" Do you know how to expand this as a sequence of partial sums? When I did the first one, I got 2 ln 2. After that I got confused.

Answer 10

But can't we transform it into \(\huge\int\large\dfrac{n+1}n\ln\left(\dfrac{n+1}n\right)dx\)?

Answer 11

ohh now i get it too thanks! :D ya it'll lead down to integration , solving \[a+\Delta i= 1+\frac{ i }{ n}, a=1, b=2\]

Answer 12

\[\int\limits_{1}^{2}x lnx dx\]

Answer 13

I have never seen this, did you just turn a sum into an integral? I'm gonna have to check this out, this looks pretty cool.

Answer 14

My you're a sharp one.

Answer 15

yeah.. its kinda crazy actually, so u mean this is the new syllabus of calculus? my generation sucks.

Answer 16

Wait my integration is screwed up lol

Answer 17

"I have never seen this, did you just turn a sum into an integral? I'm gonna have to check this out, this looks pretty cool."

Isn't that what an integral is, really? A sum, with infinitesimally small portions (dx).

Answer 18

For \[\Large \int\limits\limits_{1}^{2}x lnx dx\] this you'll prob have to do it by parts.

let u = lnx so u' = 1/x
let v' = x so v = x^2/2

Answer 19

http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts_files/eq0010MP.gif

Answer 20

okay what i dont understand is that we got f(x)=(1+x)ln(1+x) from the 1st equation.. i dont know how it came to xlnx tbh..

Answer 21

\[\Delta \sum_{i=1}^{n}=(1+i \Delta)(1+i \Delta) \] it says f(x)=(1+x)ln(1+x) in interval [0,1] OR EQUIVALENTLY, for function xlnx at [1,2]

Answer 22

actually\[1+\frac{i}{n}=\frac{n+1}{n}\]this is the c sub i value in\[\lim_{n \rightarrow \infty}\sum_{i=1}^{n}f(c _{i}) \Delta x\]

Answer 23

it says f(x)=(1+x)ln(1+x) in interval [0,1] OR EQUIVALENTLY, for function xlnx at [1,2]
when x=0, 1+x is 1.
when x=1, 1+x is 2.

They shifted the domain.

Answer 24

darn it
I meant\[1+\frac{i}{n}=\frac{n+i}{n}\]

Answer 25

oh so does that means they wanted us to be clever by shifting the domains and not going through a very tedious integration of f(x)=(1+x)ln(1+x)? whaaa T^T die alr..

Answer 26

"whaaa T^T die alr.." i've no idea what this means, but sure, that :P

Answer 27

@agent0smith Yeah of course an integral is a sum, but turning a discrete sum into an integral is quite a bit different. The closest thing I've seen is using an integral to test for convergence/divergence.

Answer 28

Ah, i see your point.

Answer 29

this is one of the questions the last year calculus exam..i am hoping for just passing this thing.. T^T stressed. oh "die already" means "i'm going to die" i live in singapore so the english here is quite crazy haha xD

Answer 30

My practice of Riemann sums always started with an integral and became a sum. It was quite a trick to start with the sum and find the patterns that would lead back to the integral. I need to go find some practice problems to get the method down.

Answer 31

i see, for me we first studied riemann then integration, the crazy part is when u actually apply riemann in integration applications problem, like this

Answer 32

Kind of a morbid question...

Most likely spot would be the maximum value of the function.

Total mass... is more difficult.

Answer 33

That is an intense question! Realistic to the point of being scary.

Answer 34

Yeah first differentiate, then turn around and integrate using only Riemann sums... better you than me :)

Answer 35

if u guys are interested, here is the solution to that

Answer 36

if i encouter such question i think ill prbly just skip haha

Answer 37

Thank you, and I understand.