I can’t conceptually grasp this the same why I can grasp addition/subtraction (the opposite of adding stuff is to take it away), but if the derivative of a function is the slope at any given point, then why would the area under the curve be the opposite of that?
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I think people forget two things: (1) that the integral is a continuous analog of a sum (it’s a long S), and (2) it’s not taking the sum of a function f(x), but the sum of f(x) times some really small value dx. Think of f(x) as a height and dx as a width of a very skinny rectangle. The integral then is the sum of the areas of these skinny rectangles, which is the total area under the curve. And at any value x, that’s the amount of area added to the total area.
Now, the derivative of the integral is the rate of change this total area over a very tiny change in x. That rate of change is the area of the skinny rectangle at x. That area is f(x)dx. Divide that over the very tiny change of x, and you get f(x)dx/dx=f(x). That’s the original function.
You can also think of f(x)dx as the total differential (total incremental amount) in the area under the curve that is added on at x. So at each x, the rate at which area is added is f(x). Generally I find differentials more intuitive than derivatives when thinking about this. The derivative isn’t really the slope, but a rate of change in something. The derivative of the integral is the rate of change in the integral, which is the rate of change in the area, which is the height of the f(x)dx rectangle at x.
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