Finding a definite integral from the Riemann sum

In summary, the conversation discusses an interval with given values, and using those values, the expression (1 + i/n)(1/n)ln[(n + i)/n] is rearranged using logarithmic properties. The question of what goes in the logarithmic function is raised, and it is suggested to simplify the argument of the logarithm. Eventually, it is realized that the argument can be simplified to (1 + i/n), resulting in the function f(x) = xlnx + (1/n)ln(1+i/n).
  • #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
1679251462458.png


Hi! I am having trouble finalizing this problem.

The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n.
Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n].
I can guess that the function is going to look something like this: f(x) = xln(...) but I don't know what goes in the logarithmic function...

I have been trying to rewrite it in terms of xiR but no luck :(
any help would be really appreciated. thank you !
 
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  • #2
Can you simplify the argument of the logarithm any further?
 
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Likes crememars
  • #3
pasmith said:
Can you simplify the argument of the logarithm any further?
Yes, I realized I was overcomplicating things. If I simplify (n+i)/n to (n/n +i/n) to (1+ i/n), I get f(x) = xlnx
 

FAQ: Finding a definite integral from the Riemann sum

What is a Riemann sum?

A Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as the integral of a function. It works by dividing the region into small sub-intervals, calculating the area of rectangles within each sub-interval, and summing these areas. The accuracy of the approximation improves as the width of these sub-intervals decreases.

How do you convert a Riemann sum into a definite integral?

To convert a Riemann sum into a definite integral, you take the limit of the Riemann sum as the number of sub-intervals approaches infinity and the width of each sub-interval approaches zero. This process effectively sums an infinite number of infinitesimally small areas, which gives you the exact area under the curve. Mathematically, this is expressed as the integral of the function over the given interval.

What is the formula for a Riemann sum?

The formula for a Riemann sum is given by: \( \sum_{i=1}^{n} f(x_i^*) \Delta x \), where \( f(x_i^*) \) is the function evaluated at a sample point \( x_i^* \) within the i-th sub-interval, \( \Delta x \) is the width of each sub-interval, and \( n \) is the number of sub-intervals. The sample point \( x_i^* \) can be chosen in various ways, such as the left endpoint, right endpoint, or midpoint of each sub-interval.

What are the different types of Riemann sums?

There are several types of Riemann sums, depending on the choice of sample points within each sub-interval. The most common types are:- Left Riemann Sum: The function is evaluated at the left endpoint of each sub-interval.- Right Riemann Sum: The function is evaluated at the right endpoint of each sub-interval.- Midpoint Riemann Sum: The function is evaluated at the midpoint of each sub-interval.- Trapezoidal Sum: The average of the left and right endpoints is used, effectively averaging the left and right Riemann sums.

Why are Riemann sums important in calculus?

Riemann sums are fundamental in calculus because they provide a way to understand and compute integrals, which are essential for measuring areas, volumes, and other quantities that accumulate over a continuous range. They form the basis for the definition of the definite integral and are a key concept in understanding the Fundamental Theorem of Calculus, which links differentiation and integration.

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