Identifying variables from Riemann sum limits

In summary, the conversation is about finding the original form of an expanded Riemann sum and determining the values for a, b, and f. The formula for calculating the sum of squares is also mentioned. The value of ∆x is discussed, with a suggested guess of 2/n. The general form of the terms is guessed to be f(x) = x^2 + 1, and it is related to f(a + i∆x) where a = 0 and b = 2.
  • #1
crememars
15
2
Homework Statement
Consider the following limit of a Riemann sum for a function f on [a, b]. Identify f, a, and b,
and express the limit as a definite integral.

*see actual expression in the description below. it was too complicated to type out so I included a picture instead.
Relevant Equations
∆x = (b-a)/n
xiR = a + i∆x
xiL = a + (i-1)∆x
1678753965145.png

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f.

If anyone could suggest how to proceed I would really appreciate it. Thank you :)
 
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  • #2
The last term is 2. For the other sums you shall use the formula
[tex]1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}[/tex]
 
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  • #3
crememars said:
Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f.

If anyone could suggest how to proceed I would really appreciate it. Thank you :)

Compare the sum to [tex]
\Delta x \sum_{i=1}^{n} f(a + i\Delta x).[/tex] What would be a good guess for [itex]\Delta x[/itex]? One of the terms is given explicitly as [tex]
\frac{4(n^2 - 2n + 1)}{n^2} =\frac{4(n-1)^2}{n^2}.[/tex] Can you guess the general form of the terms, and is your guess consistent with the first few terms given? How would you relate that general form to [itex]f(a + i\Delta x)[/itex]?
 
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  • #4
forgot to answer ! thank you for your help @pasmith and @anuttarasammyak :) I separated the terms and got f(x)= x^2 + 1

xiR = 2i/n -> x^2
n is just the riemann sum of 1 -> +1
∆x = 2/n

xiR = a + i∆x = a + 2i/n = 2i/n so a = 0 and b = 2
 

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