Fundamental Theorem of Calculus

For an integrable function $f$,

$$\int_a^b f(x) dx = F(b) - F(a),$$

where $F' = f$.

Aka the Newton-Leibnitz formula.

Interpretation

Suppose we have a flow in a piece of a pipe, $R = [a,b]$. To compute how much liquid leaves the piece we can:

• compute the out-flow/out-flow at each point and then add (integrate) the result, or simply
• compute the net amount of liquid that flows through the end points.

The result is the same. In other words:

$$\int\int_{[a,b]} F' dx = F(b) - F(a),$$

The Fundamental Theorem of Calculus is the Stokes theorem for dimension $1$.