Question

Can someone please explain how the antiderivative gives the area under a function? I don't understand logically how it works!

Answer 1

For example, if the height of the function were 3, then, for a moment, the area under the function is increasing by 3 for every 1 unit of distance you slide to the right. Keep in mind that the function can move up and down as much as it wants. As far as the function “knows”, at any particular moment it may as well be constant

So if the height of the function (which is just the function) is the rate at which the area changes, then f is the derivative of the area: A’=f. But that’s exactly the same as saying that the area is the anti-derivative of the function.

Answer 2

The formal proof is provided below.

Answer 3

Focus of the area of rectangles ABCD, ABEF.
The area of ABCD < area under curve between x and x+h < ABEF
Area ABCD < A(x+h) - A(x) < ABEF