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RobertSn
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.
The graph shown is y=x I just dont see how those steps were derived
And this is part of it also
Is therom 5 something that must be memorized? Or is there any logic to it?
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What does k represent?
k is from 1 to n, if you notice the summation sign in your first photo
But why is that over n the height?
And is the width 1/n just because we are taking it as n approaches infinity? so therefor 1/n is approach infinitely small?
yes, and sorry I'm not too familiarized in this topic... furthermore i'm busy :'(
Thanks for the input anyways kc!
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=(1-1)/n the next number is a_2=(2-1)/n=1/n <---this is how for we are away from 0 or a_1 since a_1 is 0 a_3=(3-1)/n .... now going to the kth interval where we have the x-value being a_k=(k-1)/n ... going all the way to the nth interval we have a_n=(n-1)/n because we are plugging in the left end point of that interval and not the right endpoint because we are doing left-endpoint rule we are taking all the left endpoints of all the n intervals beginning at 0
Now to find the heights of the rectangles we do f(a_k)
By a_1 does that mean a/1?
|dw:1388474479972:dw|
a_1 usually means a subscript 1
the base of each rectangle is 1/n
Alright, okay so f(k_n) would be f(k-1/n)
and thats the height at any given k
but since this function is f(x)=x then f((k-1)/n)=(k-1)/n
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,
Ok, so looking at this example, can you tell me what puzzles you and I will see if I can explain it?
Well first of all, do we know that it is left hand because of the n-1 on top of the sigma?
If that is so, for left handed questions like this can we always do (1/n)f(k/n)?
So we just set that equal to the function that is given?
yep they started at 0 and when to n-1 so left endpoint
And what does the k=xn really represent?
We are trying to find what f(x) is
\[\int\limits_{a}^{b}f(x) dx=\sum_{i=0}^{n-1} \Delta x f(a+i \cdot \Delta x)\]
where delta x =(b-a)/n
Right, and Xi= a+ i deltax/n
the easiest thing to do is assume a equals 0 so we can just look at i*delta x
Oh i see, and how about the 1?
well they chose the base to be 1/n which is delta x
if we have (b-0)/n=1/n then b has to be?
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
we can take this same problem and go a different way though
I still want to choose a to be 0 because yeah that is just plain easiest!
but with if we choose the intervals to have width 2/n instead of 1/n
then b would have to be what?
So are you also saying that you can choose some of the conventions, as long as they match up?
yes so but now we have \[\frac{2}{n} f(2 \cdot \frac{k}{n}) \text{ instead of } \frac{1}{n} f(\frac{k}{n}) \]
like are answer will look different but it will still hold the same value
\[\text{ so we want } \frac{2}{n} f(\frac{2k}{n})=\frac{1}{k+5n} \arctan(\frac{k+2n}{k+n})\]
Solve for f(2k/n) by multiplying both sides by n/2
\[f(\frac{2k }{n})=\frac{n}{2k+10n} \arctan(\frac{k+2n}{k+n})\]
Okay yea , and then how do you deal with the 2k/n?
Now we want to know what f(x) is right?
so what is k if 2k/n=x
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
Okay so we bring in an X ?
because we are trying to find out what integral notation looks like for this summation notation
looks good so we will replace all the k's we see with that
\[f(x)=\frac{n}{xn+10n}\arctan(\frac{\frac{xn}{2}+2n}{\frac{xn}{2}+n})\] \[f(x)=\frac{n(1)}{n(x+10)}\arctan(\frac{xn+4n}{xn+2n})=\frac{1}{x+10}\arctan(\frac{n(x+4)}{n(x+2)})\] \[f(x)=\frac{1}{x+10}\arctan(\frac{x+4}{x+2})\] 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
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
basically you get to choose a and b
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
This is just some applications to the definition of reimann sums let me see if i can find you some problems one moment
http://freedom.mysdhc.org/teacher/1541derflingerk/documents/Calculus/FOV1-001B4535/Integration%20via%20Sigma(2).pdf these don't look at hard but you can try these 4.3 9-12
they choose a and b for you
there c_i represents a+i*delta x by the way
If you get bored of those attempt this one I made up: Write this as an integral: \[\lim_{n \rightarrow \infty} \sum_{i=0}^{n-1} \frac{i}{i+8n} \cos(\frac{i+3n}{n})\]
Try practicing on http://www.saab.org/calculus.cgi under Select What Kind Of Problems select Calculus I( Integrals(Substitution, FTC)