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RobertSn
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Hey guys, I am looking for some help with calculus. It is in regard to rieman sums, and I would really appreciate any help understanding. I will post links below and explain further.
 7 months ago
 7 months ago
RobertSn Group Title
Hey guys, I am looking for some help with calculus. It is in regard to rieman sums, and I would really appreciate any help understanding. I will post links below and explain further.
 7 months ago
 7 months ago

This Question is Closed

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
The graph shown is y=x I just dont see how those steps were derived
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
And this is part of it also
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Is therom 5 something that must be memorized? Or is there any logic to it?
 7 months ago

kc_kennylau Group TitleBest ResponseYou've already chosen the best response.0
dw:1388473083421:dw
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
What does k represent?
 7 months ago

kc_kennylau Group TitleBest ResponseYou've already chosen the best response.0
k is from 1 to n, if you notice the summation sign in your first photo
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
But why is that over n the height?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
And is the width 1/n just because we are taking it as n approaches infinity? so therefor 1/n is approach infinitely small?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
@kc_kennylau ?
 7 months ago

kc_kennylau Group TitleBest ResponseYou've already chosen the best response.0
yes, and sorry I'm not too familiarized in this topic... furthermore i'm busy :'(
 7 months ago

kc_kennylau Group TitleBest ResponseYou've already chosen the best response.0
sorry :'(
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Thanks for the input anyways kc!
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
Ok I can try drawing a picture for you and posting it. But first let's see if you can see the pattern first. so the very first number is a_1=(11)/n the next number is a_2=(21)/n=1/n <this is how for we are away from 0 or a_1 since a_1 is 0 a_3=(31)/n .... now going to the kth interval where we have the xvalue being a_k=(k1)/n ... going all the way to the nth interval we have a_n=(n1)/n because we are plugging in the left end point of that interval and not the right endpoint because we are doing leftendpoint rule we are taking all the left endpoints of all the n intervals beginning at 0
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
Now to find the heights of the rectangles we do f(a_k)
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
By a_1 does that mean a/1?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
dw:1388474479972:dw
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
a_1 usually means a subscript 1
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
the base of each rectangle is 1/n
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Alright, okay so f(k_n) would be f(k1/n)
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
and thats the height at any given k
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
you mean f(a_k)?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
If so yes
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
but since this function is f(x)=x then f((k1)/n)=(k1)/n
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Alright I think it is starting to make sense, but when I try an example I still just dont exactly see it. Ill post if you could take a look,
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
Ok, so looking at this example, can you tell me what puzzles you and I will see if I can explain it?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Sure just a moment
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Well first of all, do we know that it is left hand because of the n1 on top of the sigma?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
If that is so, for left handed questions like this can we always do (1/n)f(k/n)?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
So we just set that equal to the function that is given?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
yep they started at 0 and when to n1 so left endpoint
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
And what does the k=xn really represent?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
We are trying to find what f(x) is
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
\[\int\limits_{a}^{b}f(x) dx=\sum_{i=0}^{n1} \Delta x f(a+i \cdot \Delta x)\]
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
where delta x =(ba)/n
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Right, and Xi= a+ i deltax/n
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
the easiest thing to do is assume a equals 0 so we can just look at i*delta x
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Oh i see, and how about the 1?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
well they chose the base to be 1/n which is delta x
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
if we have (b0)/n=1/n then b has to be?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
now this is one way to do the problem you don't have to choose it this way and the integral will still have the same value and yes b=1 for this example
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
we can take this same problem and go a different way though
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
would you like to?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Yes sure!
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
I still want to choose a to be 0 because yeah that is just plain easiest!
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
but with if we choose the intervals to have width 2/n instead of 1/n
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
then b would have to be what?
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
then b would be two
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
So are you also saying that you can choose some of the conventions, as long as they match up?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
yes so but now we have \[\frac{2}{n} f(2 \cdot \frac{k}{n}) \text{ instead of } \frac{1}{n} f(\frac{k}{n}) \]
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
like are answer will look different but it will still hold the same value
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
\[\text{ so we want } \frac{2}{n} f(\frac{2k}{n})=\frac{1}{k+5n} \arctan(\frac{k+2n}{k+n})\]
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
Solve for f(2k/n) by multiplying both sides by n/2
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
son/2 (k +5n)?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
\[f(\frac{2k }{n})=\frac{n}{2k+10n} \arctan(\frac{k+2n}{k+n})\]
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Okay yea , and then how do you deal with the 2k/n?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
Now we want to know what f(x) is right?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
so what is k if 2k/n=x
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
basically solve that for k so we can figure out what to replace k with so that we will just have x inside and not that other crap
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Okay so we bring in an X ?
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
because we are trying to find out what integral notation looks like for this summation notation
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
looks good so we will replace all the k's we see with that
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
\[f(x)=\frac{n}{xn+10n}\arctan(\frac{\frac{xn}{2}+2n}{\frac{xn}{2}+n})\] \[f(x)=\frac{n(1)}{n(x+10)}\arctan(\frac{xn+4n}{xn+2n})=\frac{1}{x+10}\arctan(\frac{n(x+4)}{n(x+2)})\] \[f(x)=\frac{1}{x+10}\arctan(\frac{x+4}{x+2})\] so this is what our f(x) looks like from choosing a= 0 and b=2 there isn't a unique answer to your question the answer can totally vary
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
http://www.wolframalpha.com/input/?i=integrate%281%2F%28x%2B10%29*arctan%28%28x%2B4%29%2F%28x%2B2%29%29%2C+x%3D0..2%29 http://www.wolframalpha.com/input/?i=integrate%281%2F%28x%2B5%29*arctan%28%28x%2B2%29%2F%28x%2B1%29%29%2C+x%3D0..1%29 see the answers look different but hold the same value
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
basically you get to choose a and b
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
and nothing else
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Wow great. Thanks a lot for your help! Do you know the exact names of these types of problems? Im looking to find some practice problems
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
This is just some applications to the definition of reimann sums let me see if i can find you some problems one moment
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
http://freedom.mysdhc.org/teacher/1541derflingerk/documents/Calculus/FOV1001B4535/Integration%20via%20Sigma(2).pdf these don't look at hard but you can try these 4.3 912
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
they choose a and b for you
 7 months ago

RobertSn Group TitleBest ResponseYou've already chosen the best response.1
Alright, thanks!
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
there c_i represents a+i*delta x by the way
 7 months ago

myininaya Group TitleBest ResponseYou've already chosen the best response.2
If you get bored of those attempt this one I made up: Write this as an integral: \[\lim_{n \rightarrow \infty} \sum_{i=0}^{n1} \frac{i}{i+8n} \cos(\frac{i+3n}{n})\]
 7 months ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.0
Try practicing on http://www.saab.org/calculus.cgi under Select What Kind Of Problems select Calculus I( Integrals(Substitution, FTC)
 7 months ago
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