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 one year ago
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.
 one year ago
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.

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

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1And this is part of it also

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

kc_kennylau
 one year ago
Best ResponseYou've already chosen the best response.0dw:1388473083421:dw

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1What does k represent?

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

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1But why is that over n the height?

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

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

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Thanks for the input anyways kc!

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2Ok 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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2Now to find the heights of the rectangles we do f(a_k)

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1By a_1 does that mean a/1?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2dw:1388474479972:dw

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2a_1 usually means a subscript 1

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2the base of each rectangle is 1/n

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Alright, okay so f(k_n) would be f(k1/n)

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1and thats the height at any given k

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2but since this function is f(x)=x then f((k1)/n)=(k1)/n

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Alright 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,

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

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

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

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1So we just set that equal to the function that is given?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2yep they started at 0 and when to n1 so left endpoint

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1And what does the k=xn really represent?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2We are trying to find what f(x) is

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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2where delta x =(ba)/n

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Right, and Xi= a+ i deltax/n

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

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Oh i see, and how about the 1?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2well they chose the base to be 1/n which is delta x

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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2now 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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2we can take this same problem and go a different way though

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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2but with if we choose the intervals to have width 2/n instead of 1/n

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2then b would have to be what?

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

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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2like are answer will look different but it will still hold the same value

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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2Solve for f(2k/n) by multiplying both sides by n/2

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2\[f(\frac{2k }{n})=\frac{n}{2k+10n} \arctan(\frac{k+2n}{k+n})\]

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Okay yea , and then how do you deal with the 2k/n?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2Now we want to know what f(x) is right?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2so what is k if 2k/n=x

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2basically 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

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Okay so we bring in an X ?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2because we are trying to find out what integral notation looks like for this summation notation

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2looks good so we will replace all the k's we see with that

myininaya
 one year ago
Best 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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2http://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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2basically you get to choose a and b

RobertSn
 one year ago
Best ResponseYou've already chosen the best response.1Wow 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

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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2http://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

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2they choose a and b for you

myininaya
 one year ago
Best ResponseYou've already chosen the best response.2there c_i represents a+i*delta x by the way

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

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