A community for students.
Here's the question you clicked on:
 0 viewing
RobertSn
 2 years 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.
RobertSn
 2 years 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.

This Question is Closed

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

RobertSn
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1388473083421:dw

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

kc_kennylau
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.1But why is that over n the height?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

myininaya
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.1Oh i see, and how about the 1?

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

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

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

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

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

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

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

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

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

myininaya
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.1Okay so we bring in an X ?

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

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

RobertSn
 2 years 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
 2 years 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
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.2they choose a and b for you

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

myininaya
 2 years 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
 2 years 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)
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.