Area under Curve

Theorem
Let $f: \R \to \R$ be a real function which is defined and (Darboux) integrable on the closed interval $\closedint a b$.

Consider the graph of $f$ embedded in a Cartesian plane.

Let $\AA$ denote the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.

Then:
 * $\AA = \ds \int_a^b \map f x \rd x$

where $\ds \int_a^b \map f x \rd x$ denotes the (Darboux) definite integral of $f$ from $a$ to $b$.

Proof

 * Area-under-Curve.png

Let $x \in \closedint a b$.

Let $\delta x$ be an arbitrarily small positive real number such that $x + \delta x \in \closedint a b$.

Consider the small strip between:
 * the $x$-axis
 * the vertical straight lines through $\tuple {x, 0}$ and $\tuple {x + \delta x, 0}$
 * the curve $\map f x$.

The area of this strip is approximated by $y \delta x$.

Hence we create a model of the geometric interpretation of the definite integral.