This is an illustration of the first fundamental theorem of calculus. The graph on the left shows rectangles [math]\Delta x[/math] wide by [math]f(x)[/math] high approximating the area under the curve at [math]x[/math]. The graph on the right is scaled to show only the additional area from [math]x[/math] to [math] x + \Delta x[/math]. The slider can be used to decrease the rectangle width and the unlabeled blue point can change the location of [math]x[/math]. 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 [math]x[/math] ). 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 [math]x[/math] increases from [math]x[/math] to [math] x + \Delta x [/math] the area under the curve increases by [math] f(x) \Delta x [/math] therefor [math] f(x) = \frac{\Delta Area}{\Delta x} [/math] or taking the limit as [math] \Delta x \rightarrow 0[/math] gives [math]Area' = f(x)[/math]. 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 [math]\Delta x[/math] decreases. What happens to the number of rectangles between A and B as [math]\Delta x[/math] decreases.