# The Fundamental Theorem of Calculus

- Author:
- Greg Petrics

The
This applet helps you explore this very important mathematical theorem.
On the left is the definite integral of

**Fundamental Theorem of Calculus**says that to calculate a definite integral of*f*(*x*), you can use any antiderivative,*F*(*x*), of*f*(*x*). Specifically:*f*(*x*) from*a*to*b*shaded red. On the right is an antiderivative,*F*(*x*), of*f*(*x*), and the difference*F*(*b*)-*F*(*a*), also shaded red. 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*are**equal*. You can explore by making changes in the app. In the left pane you can change:*f*(*x*)*a**b*

*c*, a constant of integration added to*F*(*x*). No matter what change you make, the red area on the left, and the length of the red line on the right remain equal. This is precisely the content of the Fundamental Theorem of Calculus.