What Happens When We Modify the Riemann Integral Formula?

[Karlisbad]

If we define the Riemann Integral so:

$$\sum_{i=0}^{\infty}f(X_i )(X_{i+1}-X_{i}$$

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

$$\sum_{i=0}^{\infty}f(X_i )(X_{i+1}+X_{i})0.5$$ (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:

$$\sum_{i=0}^{\infty}f(X_i )(X_{i+1}-X_{i}$$

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.