Integrals

[exchemist]

All of the functions generated by calculus via differentiation or Integration do have a small error, I do say that neither are Perfectly correct at every point, have you ever heard of holes in derivatives of functions.

 

No, but I'd like you to tell me, using my example, where the error or hole arises.

 

I am aware that functions with discontinuities have points at which the derivative or integral may be undefined, but that is something else and not general. A parabola and a straight have no such discontinuities. So let's stick to my example. 

Edited November 5, 2017 by exchemist

[DrKrettin]

All of the functions generated by calculus via differentiation or Integration do have a small error,

 

 

 

That is a strange assertion. The area of a circle is derived from integration - where is the error?

[Vmedvil]

That is a strange assertion. The area of a circle is derived from integration - where is the error?

 

Well, the sections are not small enough to account for every point in the circle the limits are equivalent to the length of the Riemann summation, if it is not infinity then n does not go to zero which causes error, so from a to b has error as n does not go to infinity as b doesn't either.

 

For a finite-sized domain, if the maximum size of a partition element shrinks to zero, this implies the number of partition elements goes to infinity. For finite partitions, Riemann sums are always approximations to the limiting value and this approximation gets better as the partition gets finer. This demonstrates how increasing the number of partitions (while lowering the maximum partition element size) better approximates the "area" under the curve:

 

I am only saying there is error in definate integrals meaning from a to b, if b = infinity, then there is no error as the partition size does actually equal zero.

Edited November 11, 2017 by Vmedvil

[exchemist]

Well, the sections are not small enough to account for every point in the circle the limits are equivalent to the length of the Riemann summation, if it is not infinity then n does not go to zero which causes error, so from a to b has error as n does not go to infinity as b doesn't either.

 

For a finite-sized domain, if the maximum size of a partition element shrinks to zero, this implies the number of partition elements goes to infinity. For finite partitions, Riemann sums are always approximations to the limiting value and this approximation gets better as the partition gets finer. This demonstrates how increasing the number of partitions (while lowering the maximum partition element size) better approximates the "area" under the curve:

 

I am only saying there is error in definate integrals meaning from a to b, if b = infinity, then there is no error as the partition size does actually equal zero.

Aha at last I think I understand your misunderstanding  here. 

 

You are confusing the "limits" of a definite integral , that is, the upper and lower bounds of the section of the function being integrated,  with the concept of a "limit" in the sense of the limit of a series.  

 

Any integral, whether definite or indefinite, represents the limit (in the sense of an infinite series) of a sum in which the width, δx, of sections of area under the curve of the function tends to zero. "dx" is used to denote the condition in which  δx -> 0. The error in the Riemann sum tends to zero as δx -> 0.  So there is no error in the integral itself: it represents the limit at which the error has vanished.