What Happens When We Modify the Riemann Integral Formula?

[Karlisbad]

If we define the Riemann Integral so:

[tex] \sum_{i=0}^{\infty}f(X_i )(X_{i+1}-X_{i} [/tex]

as a "play" what would happen if i define the integral so:

[tex] \sum_{i=0}^{\infty}f(X_i )(X_{i+1}+X_{i})0.5 [/tex] (2)

In the second definition we define the "mean value" of 2 consecutive points instead of the difference, the question is if 2 is related to the Riemann integral by some formula.

P.D:= Do Bernoulli Polynomials exist in more than 1 dimension?..

 

[matt grime]

If we define the Riemann Integral so:

[tex] \sum_{i=0}^{\infty}f(X_i )(X_{i+1}-X_{i} [/tex]

No, we do not define the Riemann integral as that.

If you were to put down what the Riemann integral really is, you'd see that you're idea above would calculate f/2.

 

[tacman]

The integral is a mathematical concept that represents the calculation of the area under a curve. It is commonly used in calculus to find the total value or quantity of a function over a given interval. The Riemann Integral is one method of calculating the integral, where the function is divided into smaller intervals and the area of each interval is approximated using a series of rectangles. The sum of these areas gives an approximation of the total area under the curve.

In the first definition provided, the Riemann Integral is defined as the sum of f(X_i) multiplied by the difference between two consecutive points, X_{i+1} and X_i. This can be represented by the formula \sum_{i=0}^{\infty}f(X_i)(X_{i+1}-X_i).

However, in the second definition, instead of using the difference between two points, the mean value of those two points is used. This can be represented by the formula \sum_{i=0}^{\infty}f(X_i)(X_{i+1}+X_i)0.5. This definition still calculates the area under the curve, but it may not be equivalent to the Riemann Integral.

To answer the question of whether 2 is related to the Riemann Integral by some formula, it depends on the specific function being integrated and the interval of integration. In general, the two definitions may give different results and are not necessarily related by a simple formula.

As for the additional question about Bernoulli Polynomials, they do exist in more than one dimension. In fact, they can be defined for any number of dimensions. However, their properties and uses may differ in higher dimensions compared to one dimension.