Area and integral

Definition: Consider the function $f(x)$ on the interval $[a,b]$.

• The function $f(x)$ is non-negative on the interval $[a,b]$ if $f(x)\geq 0$ for every $x\in[a,b]$;
• The function $f(x)$ is non-positive on the interval $[a,b]$ if $f(x)\leq 0$ for every $x\in[a,b]$.

non-negative function	non-positive function

We can relate the notion of an integral to the area of a region below the graph of a function.

Theorem:

• If $f(x)$ is a non-negative function, then the area of the region enclosed by the graph of the function $f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, is equal to the integral of $f(x)$ over the interval $[a,b]$, $O(f,a,b)=\int_a^b f(x)dx$;
• If $f(x)$ is a non-positive function, then the area of the region enclosed by the graph of the function $f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, is equal to minus the integral of $f(x)$ over the interval $[a,b]$, $O(f,a,b)=-\int_a^b f(x)dx$.

Remark: For a function that is partly non-negative and party non-positive we need to split the intervals of integration up in such a way that on each subinterval the function is either non-negative or non-positive.