Definite Integral is Area

Theorem
Let $f : \closedint a b \to \R$ be Darboux integrable over $\closedint a b$.

Let $A$ be the total signed area between $\map f x$ and the $x$-axis, and between the lines $x = a$ and $x = b$.

Then $A$ equals the Darboux integral of $f$ over $\closedint a b$.

Lemma
Define $f^+$ and $f^-$ to be the positive and negative parts of $f$, respectively.

Let $A^+$ be the area under $\map {f^+} x$ on $\closedint a b$.

Let $A^-$ be the area under $\map {f^-} x$ on the same interval.

By the lemma, $A^+$ and $A^-$ are equal to the Darboux integral over their respective parts.

Therefore: