Fundamental theorem of calculus proof

The area of the graph representing A(x + h) -  A(x) is the thin strip, we can use calculus to approximate. The top of the strip is given by the graph of a continuous function so that the values of the function do not vary very much in the little interval.

It follows that the thin strip is approximately a rectangle and its area can be estimated by multiplying the width by an estimate for the height.

The width is simply (x + h) - x = h.

The height can be estimated by picking any sample point o in the interval [x, x+h] and using f(p) for a representative height. Thus for p any value in the interval [x,x+h],

A(x + h) - A(x) approximately equals .

Looking again at the equation:

you see that This equation  is approximately equals .

Finally, as h --> 0, the point p, which is sandwiched in between x and x+h, gets closer and closer to x, and again by continuity of f, the values f(p) get closer and closer to the value of f(x). Thus under the assumption that the function f is continuous on the interval [a,b], it is true that:

This concludes the informal proof of the fundamental theorem of calculus.