The [b]Fundamental Theorem of Calculus[/b] says that to calculate a definite integral of [i]f[/i]([i]x[/i]), you can use any antiderivative, [i]F[/i]([i]x[/i]), of [i]f[/i]([i]x[/i]). Specifically:[br][br][math]\int_a^bf\left(x\right)dx=F\left(b\right)-F\left(a\right)[/math][br][br]This applet helps you explore this very important mathematical theorem.[br][br]On the left is the definite integral of [i]f[/i]([i]x[/i]) from [i]a[/i] to [i]b[/i] shaded [color=#ff0000]red[/color].[br][br]On the right is an antiderivative, [i]F[/i]([i]x[/i]), of [i]f[/i]([i]x[/i]), and the difference [i]F[/i]([i]b[/i])-[i]F[/i]([i]a[/i]), also shaded [color=#ff0000]red[/color].[br][br]The Fundamental Theorem of Calculus (amazingly) says that these two numbers are equal. There is not an immediately obvious reason why these two numbers are equal. Nonetheless, these two numbers [i]are[/i] [i]equal[/i].[br][br]You can explore by making changes in the app.[br][br]In the left pane you can change:[br][list][*][i]f[/i]([i]x[/i])[/*][*][i]a[/i][/*][*][i]b[/i][/*][/list][br]In the right pane, you can change [i]c[/i], a constant of integration added to [i]F[/i]([i]x[/i]).[br][br]No matter what change you make, the[color=#ff0000] red area[/color] on the left, and the length of the [color=#ff0000]red line[/color] on the right remain equal. This is precisely the content of the Fundamental Theorem of Calculus.