Definition:Darboux Integral/Geometric Interpretation

Definition
Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.

Let $f: \closedint a b \to \R$ be a function.

Let $\map f x$ be bounded on $\closedint a b$.

Let:
 * $\ds \int_a^b \map f x \rd x$

be the definite (Darboux) integral of $\map f x$ over $\closedint a b$.

The expression $\ds \int_a^b \map f x \rd x$ can be (and frequently is) interpreted as the area under the graph.

This follows from the definition of the definite integral as a sum of the product of the lengths of intervals and the "height" of the function being integrated in that interval and the formula for the area of a rectangle.

A depiction of the lower and upper sums illustrates this:


 * RiemannLowerSum.png RiemannUpperSum.png

It can intuitively be seen that as the number of points in the subdivision increases, the more "accurate" the lower and upper sums become.

Also note that if the graph is below the $x$-axis, the signed area under the graph becomes negative.

Also see
This is rigorously demonstrated in Definite Integral is Area.