Solving Riemann Sum Problem: Integral of x^x

In summary, the conversation is about an attempt to solve the integral $$\int_{0}^{1} x^{x} dx$$ using Riemann sums. However, the speaker comes to the conclusion that the result is 0, which is incorrect. They discuss the mistake of expanding the sum and the difficulty of finding a closed formula for the sum. They also mention that Riemann sums are not always easy to evaluate and consider trying a different approach. The conversation ends with the speaker consulting Wikipedia for information about the series.
  • #1
MAGNIBORO
106
26
Hi.
I try to solve the integral $$\int_{0}^{1} x^{x} dx$$
Through sums of riemann But I came to the conclusion that the result is 0 that is wrong
$$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$
$$= \lim_{n\rightarrow \infty }\frac{\frac{1}{n}^{\frac{1}{n}}}{n} + \frac{\frac{2}{n}^{\frac{2}{n}}}{n} + ... +\frac{\frac{n-1}{n}^{\frac{n-1}{n}}}{n} + \frac{1}{n}=0$$
$$\int_{0}^{1} x^{x} dx = 0 $$

I'm sure the mistake is expand the ##\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}##

because in some easy integrals like ##\int_{0}^{1} x dx## if you expand the ##\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )## you get
$$= \lim_{n\rightarrow \infty } \frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n-1}{n^{2}}+\frac{1}{n} = 0$$
Instead of finding a "closed" formula like:
$$= \lim_{n\rightarrow \infty } \frac{n(n+1)}{2\, n^{2}}=\frac{1}{2}$$
so If I can not find a formula like ## \frac{n(n+1)}{2}## for ## \sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}##
the limit of the riemman sum is wrong?
 
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  • #2
Where does the first "=0" come from? The limit of each fraction is 0, but you add more terms in each step, you cannot simply look at the limits of each fraction.

The limit is wrong, but the problem is not the lack of a general formula for the sum.
 
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Likes MAGNIBORO
  • #3
mfb said:
Where does the first "=0" come from? The limit of each fraction is 0, but you add more terms in each step, you cannot simply look at the limits of each fraction.

The limit is wrong, but the problem is not the lack of a general formula for the sum.

Ok then what is the correct way to solve the problem?
Because in simple cases as in polynomials it is enough Arrive at a formula to resolve the limit.
in more complex function is "easy" to solve the problem Through riemann sums or is very difficult?
Maybe I have to change the method?
 
  • #4
What makes you think there is a nice closed form for the solution?
In general Riemann sums are not easy to evaluate. For polynomials they work, but even there they are not the easiest approach.
 
  • #5
According to wolframalpha, the series diverges!
 
  • #7
mfb said:
What makes you think there is a nice closed form for the solution?
In general Riemann sums are not easy to evaluate. For polynomials they work, but even there they are not the easiest approach.
well I suspected it but as the sum of riemann is the definition of integral I thought they might be useful for something other than polynomials
so Then I'll try another approach. thanks
ShayanJ said:
According to wolframalpha, the series diverges!
I believe you but wolframalpha hates me
https://www.wolframalpha.com/input/...rmassumption={"C",+"limit"}+->+{"Calculator"}
Stephen Tashi said:
I already knew the result but wanted to get it myself And as it is a sum I thought that the sums of riemann or the series of taylor would be a good approximation
 

FAQ: Solving Riemann Sum Problem: Integral of x^x

1. What is a Riemann sum?

A Riemann sum is a method of approximating the area under a curve by dividing it into smaller rectangles and finding the sum of their areas. As the number of rectangles increases, the approximation becomes more accurate.

2. How do you calculate a Riemann sum?

To calculate a Riemann sum, you first divide the interval into smaller subintervals. Then, you choose a sample point within each subinterval and evaluate the function at that point. Finally, you multiply the function value by the width of the subinterval and add all of these products together.

3. What is the purpose of solving Riemann sum problems?

The purpose of solving Riemann sum problems is to approximate the area under a curve when the function cannot be integrated using basic integration techniques. It is also used to understand the concept of integration and to evaluate definite integrals numerically.

4. How do you find the integral of x^x?

The integral of x^x cannot be found using basic integration techniques. Instead, you can use the Riemann sum method to approximate the integral by dividing the interval into smaller subintervals and evaluating the function at sample points within each subinterval.

5. Are there any limitations to using Riemann sums?

Yes, there are limitations to using Riemann sums. As the number of rectangles increases, the calculation becomes more complex and time-consuming. It also only provides an approximation of the actual integral and may not be completely accurate.

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