In this activity we start with a function f(x) which is graphed in the left window. Enter the desired formula in the input box for f(x). For now set [i]a[/i] = 0,[i] x[/i] = 0, and [i]C[/i] = 0 via their sliders or input boxes. Now slide the slider for [i]x[/i] slowly to the right. [br][br]The shaded area is accumulating as we increase the value for [i]x[/i]. Green areas accumulate positively and red areas accumulate negatively. The green area minus the red area is the value of the integral of [i]f[/i]([i]x[/i]) over [br]the interval [[i]a, x[/i]]. This is demonstrated by the size of the vertical line segment in the window on the right. Note that this defines a new accumulation function [i]A[/i]. Note that this function goes through the [br]origin.[br][br]Check the checkbox for [i]A[/i]([i]x[/i]) in the right window. You will now see the entire accumulation function. On this blue accumulation function which is graphed in the right window, [i]y[/i]-values are equal to the green area - [br]red area from the window on the left.[br][br]If we adjust the value of a, then the accumulation will now go through ([i]a[/i], 0) instead of the origin.[br][br]If we adjust the value of [i]C[/i], then the entire accumulation function will be shifted vertically by [i]C[/i] units.[br][br]Note that this gives us a way to take any function we can integrate and use it to define a new related accumulation function. Actually, there are a whole family of these accumulation functions for different choices of [i]a[/i] and [i]C[/i].

(Fundamental Theorem of Calculus Part 2)[br]Click on the [i]A[/i]'([i]x[/i]) checkbox in the right window. This will graph the derivative of the accumulation function in red in the right window. [br][br]How does [i]A[/i]'([i]x[/i]) compare to the original [i]f[/i]([i]x[/i])?[br][br]They are the same! This illustrates the [br][br][b]Second Fundamental Theorem of Calculus[/b][br][br]For any function [i]f[/i] which is continuous on the interval containing[i] a, x[/i], and all values between them:[br][br][math]\frac{d}{dx}\left(\int_z^xf\left(t\right)dt\right)=f\left(x\right)[/math][br][br]This tells us that each of these accumulation functions are antiderivatives of the original function [i]f[/i].[br][br]First integrating and then differentiating returns you back to the original function. In this sense, integration and differentiation are inverse processes.[br]