What is the point of the definate integral?

[grayson]

Okay, I know that the definite integral is used in finding the "AREA UNDER A CURVE" And there is also maxwell's electromagnetism equation I think. I know that there is a lot of examples, but how did they all find out how and why to use them?

Sorry, I meant indefinite

[LaurieAG]

grayson, the following is just my opinion. Also don't forget about proper and improper integrals.

Elementary Calculus an Infinitessimal Approach http://Reference link: https://www.math.wisc.edu/~keisler/keislercalc-03-07-17.pdf

https://arxiv.org/abs/physics/9807044v2

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Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as ‘the failure of the energy theorem’. In a correspondence with Klein [3], he asserted that this ‘failure’ is a characteristic feature of the general theory, and that instead of ‘proper energy theorems’ one had ‘improper energy theorems’ in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

At a conceptual structural level improper integrals in calculus are integrals, usually with limits from +infinity to -infinity, that converge at their limits. Refer H.J. Keisler, p367, Definition to P370, examples 7, 8, and 9. If they don't converge then they are indefinite integrals which are entirely different. Refer H.J. Keisler p370, example 10, diagram 6.7.10

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It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox... So we do not give the integral ... the value 0, instead leave it undefined.

That doesn't mean that indefinite integrals don't play a part in our calculus or physics as an indefinite integral that cycles between +infinity and -infinity, as a sub function of a higher level function, is a valid proper use of indefinite integrals. Refer H.J. Keisler p224-5, Definition and example 8, diagram 4.4.6 second equation with u and substitute infinite limits.

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We do not know how to find the indefinite integrals in this example. Nevertheless the answer is 0 because on changing variables both limits of integration become the same.

A valid proper integral of any form is not equivalent to a valid improper integral because that is the underlying conceptual difference between classical and modern physics as discussed by Hilbert and Klein above.

While Emmy Noether provided the conceptual symmetries of relativity Arthur Compton provided the final piece of the puzzle by experimentally and theoretically uniting the wave and particle natures of electromagnetic particles between 1922-23. He was awarded the Nobel prize in physics for his work in 1927.

 

[swansont]

15 hours ago, grayson said:

Okay, I know that the definite integral is used in finding the "AREA UNDER A CURVE" And there is also maxwell's electromagnetism equation I think. I know that there is a lot of examples, but how did they all find out how and why to use them?

Sorry, I meant indefinite

Sometimes the function is important, rather than the numerical result of a definite integral

e.g. if you integrate Fdx, rewritten as mvdv, you get 1/2 mv^2, the equation for kinetic energy

Introductory physics has many such examples

[amaila3]

Certainly! The concept of definite and indefinite integrals is fundamental in mathematics and has various real-world applications. Definite integrals, as you mentioned, are used to find the "area under a curve" and play a crucial role in calculus, physics, and engineering. For instance, in physics, they help calculate quantities like displacement, velocity, and acceleration.

Maxwell's electromagnetism equations, part of the field of electromagnetism, also rely on integral calculus for solving complex problems related to electric and magnetic fields. This shows the wide-reaching impact of integrals in understanding the behavior of our physical world.

Regarding their discovery and use, integral calculus was developed over centuries by mathematicians like Newton and Leibniz, who independently developed the fundamental theorems of calculus. They devised these concepts to solve problems in physics and mathematics.

In essence, the discovery and application of integrals are the result of centuries of mathematical development driven by the need to understand and solve real-world problems in various fields, making them indispensable tools in science and engineering today.

[jhamesguru07a]

The indefinite integral, or antiderivative, is a mathematical operation that helps find the original function from its derivative. It includes an arbitrary constant and is essential in calculus.

[studiot]

3 hours ago, jhamesguru07a said:

The indefinite integral, or antiderivative, is a mathematical operation that helps find the original function from its derivative. It includes an arbitrary constant and is essential in calculus.

I'm sorry but what has this to do with the OP ?