Is there a general way to solve integrals?

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Homework Help Overview

The discussion revolves around the concept of integrals in calculus, specifically whether there is a general method to prove integrals similar to the derivative formula. Participants explore the relationship between Riemann sums and integral computation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of Riemann sums as a method for proving integrals and question the feasibility of proving most integrals without the Fundamental Theorem of Calculus (FTC). There is also a focus on defining what "most integrals" entails, particularly in relation to ILATE functions.

Discussion Status

The conversation is ongoing, with some participants providing insights about Riemann sums and their historical context in integral computation. Questions about the applicability of this method to various types of integrals remain open for exploration.

Contextual Notes

There is a mention of the infinite nature of integrals and the distinction between those solvable in elementary functions versus others. The discussion also highlights the need for clarity in defining terms like "most integrals."

Voivode
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Homework Statement



This is just something I've been wondering, but since derivatives have the formula:

dy/dx = lim h-> 0 of (f(x+h) - f(x)/h)

And that formula can prove a lot of derivatives.

Does a similar formula exist that can prove integrals?

Homework Equations



The Attempt at a Solution

 
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Yes. Check out riemann sums.

You take the limits of areas of rectangles under the curve as the number of rectangles approaches infinity.
 
CalculusHelp1 said:
Yes. Check out riemann sums.

You take the limits of areas of rectangles under the curve as the number of rectangles approaches infinity.

This is actually the original way that integrals were computed before the discovery of the Fundamental Theorem of Calculus.
 
Would it be possible to prove most integrals without the FTC using Riemann Sums?
 
Voivode said:
Would it be possible to prove most integrals without the FTC using Riemann Sums?

Well, first you'd have to define what you mean by "most integrals". After all, there are an infinite amount of integrals with a solution in the elementary functions. I am relatively certain that all polynomial integrals can be solved in this fashion.
 
By most integrals, I was mainly thinking of the ILATE functions. Could those be solved with riemann sums?
 

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