Why is the integral of a function the area?

Say you have two functions, A and B, and A is the gradient of B. That is to say, A is as high as B is steep (at any point). Equally, A is as high as the rate of change of B (at any point). Now let’s say we want to find a function that tells us the area under the curve A. Since the area under the curve A will increase as fast as A is high (think: if A is really high, then moving even a little along in x will result in a massive change in area still), we must be looking for a function which is increasing as fast as A is high. Rephrasing this a little bit, we want a function whose gradient (rate of increase) is as high as A is, which from our definition of a derivative is exactly the function B since A is the gradient of B. Thus, to find the area under a function A, we are always looking for the function B, which when differentiated, produces A, and so is the integral of A.