What are the first principles of integration?

[Mentallic]

I don't know much about the history of how calculus was derived, but I would believe that derivatives were first invented to find the gradient of a tangent to a curve.

From first principles in calculus, it makes sense to me how the tangent is found. However, I don't understand how mathematicians knew or discovered that while the derivatives give the gradient, the anti-derivatives (or integration) give the area under the curve.

I guess what I'm asking for is if there are any first principles of integration like there is for the tangents?

 

[JG89]

https://www.physicsforums.com/showthread.php?t=324858


Check out my post in that thread, it should answer your question.

 

[HallsofIvy]

Actually, the basics of finding integrals by "exhaustion" go back to Archimedes and long predate the problem of finding slopes of tangent lines (derivatives).

There were a number of different ways of finding tangent lines at the time of Newton and Leibniz. One of the things that made Newton and Leibniz the "founders" of Calculus is that they recognized that the two problems were different aspects of the same thing- in effect "inverse" calculations.

 

[daniel_i_l]

It makes sense that summing up the changes that the function goes through from a to b [ie: integrating the derivative of a function from a to b] will give you the total change which is f(b) - f(a).

For example, if every second I take a step either forwards or backwards, then if I sum up the steps over an hour I'll get the total distance that I walked.