Express sum as a definite integral

In summary, the problem involves finding a definite integral for the expression given in the limit, where n is a positive integer. The student was unsure of how to approach the problem and asked for help, eventually realizing that it can be solved using Riemann sums as approximations to integrals. They now just need to determine the function f(x) in order to complete the solution.
  • #1
Bohrok
867
0
Homework Statement

If n is a positive integer, then
[tex]\lim_{n\to\infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^2+\left(\frac{2}{n}\right)^2+\cdot\cdot\cdot+\left(\frac{n-1}{n}\right)^2\right][/tex]can be expressed by what definite integral?

The attempt at a solution

A student I was helping had this problem and I had no idea how to even start. It was a problem along with other basic calc I definite and indefinite integrals, so I'm guessing it has some easy solution that I'm completely missing.
 
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  • #2
You don't know Riemann sums as approximations to integrals??
 
  • #3
I knew it was something simple :rolleyes:
Makes sense; now I just need to figure out f(x)...
 

Related to Express sum as a definite integral

1. What does it mean to express a sum as a definite integral?

Expressing a sum as a definite integral means representing a sum of infinitely many terms as the limit of a Riemann sum, where the intervals between each term get infinitely small.

2. Why is it useful to express a sum as a definite integral?

Expressing a sum as a definite integral allows us to evaluate the sum more efficiently by using properties of integrals, such as the Fundamental Theorem of Calculus. It also helps us to visualize the sum as a continuous function, making it easier to understand and work with.

3. How do you express a sum as a definite integral?

To express a sum as a definite integral, we first need to identify the pattern or general term of the sum. Then, we can write the sum as an infinite series and use properties of integrals to rewrite it as a definite integral. Finally, we take the limit as the number of terms approaches infinity to get the definite integral.

4. Can any sum be expressed as a definite integral?

No, not all sums can be expressed as a definite integral. The sum must have a pattern or general term that allows us to write it as an infinite series. Additionally, the sum must converge in order for us to take the limit and evaluate it as a definite integral.

5. What are some real-life applications of expressing a sum as a definite integral?

Expressing a sum as a definite integral has many real-life applications, such as calculating areas and volumes, finding the average value of a function, and solving problems in physics and engineering. It is also used in probability and statistics to calculate probabilities and expected values.

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