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
 

FAQ: Identifying variables from Riemann sum limits

What is a Riemann sum?

A Riemann sum is a method for approximating the total area under a curve on a graph, which represents an integral. It is calculated by dividing the region into small subintervals, computing the function's value at specific points within these intervals, and then summing up the products of these values and the widths of the intervals.

How do you identify the variable of integration from a Riemann sum limit?

The variable of integration is usually identified as the variable that changes within the subintervals. In a Riemann sum, it is typically represented by a symbol like 'x' or 't' and appears in the function being summed. It is also the variable with respect to which the limit of the sum is taken as the number of subintervals approaches infinity.

What is the connection between a Riemann sum and a definite integral?

A Riemann sum is an approximation of a definite integral. As the number of subintervals increases and their width decreases, the Riemann sum approaches the exact value of the definite integral. Mathematically, the limit of the Riemann sum as the number of subintervals goes to infinity is the definite integral of the function over the interval.

How do you determine the width of the subintervals in a Riemann sum?

The width of each subinterval in a Riemann sum is typically denoted by Δx (delta x) and is calculated as (b - a) / n, where [a, b] is the interval over which the integration is performed, and n is the number of subintervals. As n increases, Δx decreases, leading to a more accurate approximation of the integral.

What are the different types of Riemann sums?

There are several types of Riemann sums, including the left Riemann sum, right Riemann sum, midpoint Riemann sum, and trapezoidal sum. The left Riemann sum uses the left endpoint of each subinterval for the function value, the right Riemann sum uses the right endpoint, the midpoint sum uses the midpoint, and the trapezoidal sum uses the average of the left and right endpoints to approximate the area under the curve.

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