## 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. 8 months ago 8 months ago

1. RobertSn Group Title

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

2. RobertSn Group Title

And this is part of it also

3. RobertSn Group Title

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

4. kc_kennylau Group Title

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5. RobertSn Group Title

What does k represent?

6. kc_kennylau Group Title

7. RobertSn Group Title

But why is that over n the height?

8. RobertSn Group Title

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

9. RobertSn Group Title

@kc_kennylau ?

10. kc_kennylau Group Title

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

11. kc_kennylau Group Title

sorry :'(

12. RobertSn Group Title

Thanks for the input anyways kc!

13. myininaya Group Title

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

14. myininaya Group Title

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

15. RobertSn Group Title

By a_1 does that mean a/1?

16. myininaya Group Title

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17. myininaya Group Title

a_1 usually means a subscript 1

18. myininaya Group Title

the base of each rectangle is 1/n

19. RobertSn Group Title

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

20. RobertSn Group Title

and thats the height at any given k

21. myininaya Group Title

you mean f(a_k)?

22. myininaya Group Title

If so yes

23. myininaya Group Title

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

24. RobertSn Group Title

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,

25. myininaya Group Title

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

26. RobertSn Group Title

Sure just a moment

27. RobertSn Group Title

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

28. RobertSn Group Title

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

29. RobertSn Group Title

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

30. myininaya Group Title

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

31. RobertSn Group Title

And what does the k=xn really represent?

32. myininaya Group Title

We are trying to find what f(x) is

33. myininaya Group Title

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

34. myininaya Group Title

where delta x =(b-a)/n

35. RobertSn Group Title

Right, and Xi= a+ i deltax/n

36. myininaya Group Title

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

37. RobertSn Group Title

Oh i see, and how about the 1?

38. myininaya Group Title

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

39. myininaya Group Title

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

40. RobertSn Group Title

1

41. myininaya Group Title

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

42. myininaya Group Title

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

43. myininaya Group Title

would you like to?

44. RobertSn Group Title

Yes sure!

45. myininaya Group Title

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

46. myininaya Group Title

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

47. myininaya Group Title

then b would have to be what?

48. RobertSn Group Title

then b would be two

49. RobertSn Group Title

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

50. myininaya Group Title

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

51. myininaya Group Title

yep

52. myininaya Group Title

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

53. myininaya Group Title

our*

54. myininaya Group Title

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

55. myininaya Group Title

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

56. RobertSn Group Title

son/2 (k +5n)?

57. myininaya Group Title

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

58. RobertSn Group Title

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

59. myininaya Group Title

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

60. RobertSn Group Title

right

61. myininaya Group Title

so what is k if 2k/n=x

62. myininaya Group Title

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

63. RobertSn Group Title

Okay so we bring in an X ?

64. RobertSn Group Title

xn/2=k

65. myininaya Group Title

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

66. myininaya Group Title

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

67. myininaya Group Title

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

68. myininaya Group Title
69. myininaya Group Title

basically you get to choose a and b

70. myininaya Group Title

and nothing else

71. RobertSn Group Title

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

72. myininaya Group Title

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

73. myininaya Group Title

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

74. myininaya Group Title

as*

75. myininaya Group Title

they choose a and b for you

76. RobertSn Group Title

Alright, thanks!

77. myininaya Group Title

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

78. myininaya Group Title

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

79. eliassaab Group Title

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