Exploring the Differences Between Newton and Riemann Integrals

In summary, the difference between the Newton integral and the Riemann integral is that the Newton integral is defined as one side of the second fundamental theorem of calculus, while the Riemann integral is the other side. The fundamental theorem of calculus states that the two definitions of integral agree. The Newton integral is like a black box, but the Riemann integral is more rigorously defined and it is harder to define. To define the Newton integral, one must find a function G that satisfies the integral of f is G(b) - G(a). However, the right answer is that G can be any Lipschitz continuous function whose derivative exists and equals f almost everywhere. If such a G cannot be found, the integral can be approx
  • #1
pivoxa15
2,255
1
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?
 
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  • #2
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.
 
  • #3
That seems to make sense. The Newton integral is like a black box but really the proper one is the Riemann integral.
 
  • #4
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.
 

Related to Exploring the Differences Between Newton and Riemann Integrals

1) What is the difference between Newton and Riemann integrals?

The main difference between Newton and Riemann integrals lies in their approach to calculating the area under a curve. Newton's method involves finding the antiderivative of a function and evaluating it at the upper and lower limits of integration. Riemann's method, on the other hand, involves dividing the area under the curve into smaller rectangles and summing their areas to approximate the integral.

2) Which integral is more commonly used in real-world applications?

Riemann integrals are more commonly used in real-world applications because they can handle a wider range of functions than Newton integrals. Riemann integrals are also easier to calculate and more intuitive to understand, making them a more practical choice for most applications.

3) Can Riemann integrals be used to calculate improper integrals?

Yes, Riemann integrals can be used to calculate improper integrals. Improper integrals are integrals with infinite limits of integration or have a discontinuity within the interval of integration. Riemann's method can handle these cases by breaking up the integral into smaller intervals and summing their areas, leading to an accurate approximation of the integral.

4) What are the limitations of Newton and Riemann integrals?

One limitation of Riemann integrals is that they cannot be used to calculate integrals for functions that are not continuous or have discontinuities within the interval of integration. On the other hand, Newton integrals are limited in their applicability to functions that have an antiderivative that can be expressed in terms of elementary functions.

5) Can both Newton and Riemann integrals be used to solve definite and indefinite integrals?

Yes, both Newton and Riemann integrals can be used to solve both definite and indefinite integrals. Definite integrals have specific limits of integration, while indefinite integrals do not. Both methods can handle both types of integrals, although Riemann integrals are more commonly used for definite integrals and Newton integrals for indefinite integrals.

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