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.

- anonymous

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- anonymous

The graph shown is y=x
I just dont see how those steps were derived

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- anonymous

And this is part of it also

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- anonymous

Is therom 5 something that must be memorized? Or is there any logic to it?

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- kc_kennylau

|dw:1388473083421:dw|

- anonymous

What does k represent?

- kc_kennylau

k is from 1 to n, if you notice the summation sign in your first photo

- anonymous

But why is that over n the height?

- anonymous

And is the width 1/n just because we are taking it as n approaches infinity? so therefor 1/n is approach infinitely small?

- anonymous

@kc_kennylau ?

- kc_kennylau

yes, and sorry I'm not too familiarized in this topic... furthermore i'm busy :'(

- kc_kennylau

sorry :'(

- anonymous

Thanks for the input anyways kc!

- myininaya

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

- myininaya

Now to find the heights of the rectangles we do f(a_k)

- anonymous

By a_1 does that mean a/1?

- myininaya

|dw:1388474479972:dw|

- myininaya

a_1 usually means a subscript 1

- myininaya

the base of each rectangle is 1/n

- anonymous

Alright, okay so f(k_n) would be f(k-1/n)

- anonymous

and thats the height at any given k

- myininaya

you mean f(a_k)?

- myininaya

If so yes

- myininaya

but since this function is f(x)=x then f((k-1)/n)=(k-1)/n

- anonymous

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,

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- myininaya

Ok, so looking at this example, can you tell me what puzzles you and I will see if I can explain it?

- anonymous

Sure just a moment

- anonymous

Well first of all, do we know that it is left hand because of the n-1 on top of the sigma?

- anonymous

If that is so, for left handed questions like this can we always do (1/n)f(k/n)?

- anonymous

So we just set that equal to the function that is given?

- myininaya

yep they started at 0 and when to n-1 so left endpoint

- anonymous

And what does the k=xn really represent?

- myininaya

We are trying to find what f(x) is

- myininaya

\[\int\limits_{a}^{b}f(x) dx=\sum_{i=0}^{n-1} \Delta x f(a+i \cdot \Delta x)\]

- myininaya

where delta x =(b-a)/n

- anonymous

Right, and Xi= a+ i deltax/n

- myininaya

the easiest thing to do is assume a equals 0 so we can just look at i*delta x

- anonymous

Oh i see, and how about the 1?

- myininaya

well they chose the base to be 1/n which is delta x

- myininaya

if we have (b-0)/n=1/n then b has to be?

- anonymous

1

- myininaya

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

- myininaya

we can take this same problem and go a different way though

- myininaya

would you like to?

- anonymous

Yes sure!

- myininaya

I still want to choose a to be 0 because yeah that is just plain easiest!

- myininaya

but with if we choose the intervals to have width 2/n instead of 1/n

- myininaya

then b would have to be what?

- anonymous

then b would be two

- anonymous

So are you also saying that you can choose some of the conventions, as long as they match up?

- myininaya

yes so but now we have
\[\frac{2}{n} f(2 \cdot \frac{k}{n}) \text{ instead of } \frac{1}{n} f(\frac{k}{n}) \]

- myininaya

yep

- myininaya

like are answer will look different but it will still hold the same value

- myininaya

our*

- myininaya

\[\text{ so we want } \frac{2}{n} f(\frac{2k}{n})=\frac{1}{k+5n} \arctan(\frac{k+2n}{k+n})\]

- myininaya

Solve for f(2k/n) by multiplying both sides by n/2

- anonymous

son/2 (k +5n)?

- myininaya

\[f(\frac{2k }{n})=\frac{n}{2k+10n} \arctan(\frac{k+2n}{k+n})\]

- anonymous

Okay yea , and then how do you deal with the 2k/n?

- myininaya

Now we want to know what f(x) is right?

- anonymous

right

- myininaya

so what is k if 2k/n=x

- myininaya

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

- anonymous

Okay so we bring in an X ?

- anonymous

xn/2=k

- myininaya

because we are trying to find out what integral notation looks like for this summation notation

- myininaya

looks good so we will replace all the k's we see with that

- myininaya

\[f(x)=\frac{n}{xn+10n}\arctan(\frac{\frac{xn}{2}+2n}{\frac{xn}{2}+n})\]
\[f(x)=\frac{n(1)}{n(x+10)}\arctan(\frac{xn+4n}{xn+2n})=\frac{1}{x+10}\arctan(\frac{n(x+4)}{n(x+2)})\]
\[f(x)=\frac{1}{x+10}\arctan(\frac{x+4}{x+2})\]
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

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

- myininaya

basically you get to choose a and b

- myininaya

and nothing else

- anonymous

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

- myininaya

This is just some applications to the definition of reimann sums
let me see if i can find you some problems
one moment

- myininaya

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

- myininaya

as*

- myininaya

they choose a and b for you

- anonymous

Alright, thanks!

- myininaya

there c_i represents a+i*delta x by the way

- myininaya

If you get bored of those attempt this one I made up:
Write this as an integral:
\[\lim_{n \rightarrow \infty} \sum_{i=0}^{n-1} \frac{i}{i+8n} \cos(\frac{i+3n}{n})\]

- anonymous

Try practicing on
http://www.saab.org/calculus.cgi
under
Select What Kind Of Problems
select
Calculus I( Integrals(Substitution, FTC)

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