In the earlier explanation of area and primitive function we secretly assumed that $f(x)\ge 0$ all $x$ in $[a,b]$. But what to do when the function $f(x)$ also attains negative values?

If $f(x)\le 0$ for all $x$ in $[a,b]$, then $-f(x)\ge 0$ and the function $O_{-f}(x)$ that represents the area of the region enclosed by the $x$-axis, the vertical line through the point $(x,0)$ and graph of $-f(x)$ is a primitive function of $-f(x)$. This is illustrated in the figure below.

Because $O\,'_{-f}(x)=-f(x)$, then in turn $O(x)=-O_{-f}(x)$ is a primitive function of $f(x)$. But this is also the area on $[a,x]$ between the horizontal $x$-axis and the graph of $f(x)$ with an additional minus sign. If the area of a plane region below the horizontal axis is considered a negative quantity, then $\int_a^b f(x)\,\dd x$ still represents the area between the $x$-axis, the graph of $f(x)$ and the vertical lines through $(a,0)$ and $(b,0)$.

If $f(x)$ changes sign on $[a,b]$, positive and negative contributions must be combined. For regions where the function values are negative, the contributions to the area must be taken with a minus. In the figure below, there exist on the interval two regions with a positive contribution and one region with a negative contribution to the area.

In this interpretation of area, the following statement remains true: