Integration Area Under A Curve help

Would somebody also know how integration can find the area under a curve. I understand how the gradient lets say of a function f(x) affects the area underneath. Isn't it also that f '(x) is the rate of change of y with respect to x. ( it is the gradient of the curvy function)... So therefore f'(x), the gradient, must have to be related to the area somehow. I understand how to do the calculations, it's merely understanding how the method finds it. Any help would be very appreciated plus rep.

Has anybody anything else to help understand? Rep btw