First Fundamental Theorem of Calculus

This is an illustration of the first fundamental theorem of calculus. The graph on the left shows rectangles wide by high approximating the area under the curve at . The graph on the right is scaled to show only the additional area from to . The slider can be used to decrease the rectangle width and the unlabeled blue point can change the location of . The points A and B can be used to change the limits of integration, ( B does not really do anything since the other point controls ). The function can be modified by moving the black points.
By looking at how the area increases as each new rectangle is added a relationship for the area can be derived. As increases from to the area under the curve increases by therefor or taking the limit as gives . This implies the Area under the curve is the anti-derivative of the function of the curve. Comment on how closely the function matches the top of the rectangle as decreases. What happens to the number of rectangles between A and B as decreases.