What Are the Key Differences Between Newton and Riemann Integrals?

[pivoxa15]

What is the difference between the two? Does the Newton integral arise from the fundalmental theorem of calculus and the Riemann integral is the Newton integral but more rigorously defined?

 

[mathwonk]

briefly the Newton integral seems to be defined as one side of the second FTC and the riemann integral is the other side, so the FTC seems to say the two definitions of integral agree.

 

[pivoxa15]

That seems to make sense. The Newton integral is like a black box but really the proper one is the Riemann integral.

 

[mathwonk]

and it is haeder to define. suppsoe f is any function, and i want to define its enwton integral. i havre tot ell you how to recognize a function G such thast the integral of f is G(b) - G(a).

the right answer, if is only riemNN integrable, and not necessarily continuous, is that G is any liposchitz continuous function whosae derivative exists and equals f almost everywhere.

but how do yopum find such a G? and if you cannot find one, how do you approximate the integral?

seems hopeless without the limit of riemann sums definition.