5.3 Areas and integrals

The fundamental idea is that the area of a rectangle equals base times height. To calculate other areas, we approximate by rectangles. Let $f(x)$ be a continuous functions defined for $a\leq x\leq b$; we call $a$ and $b$ the lower and upper limits of integration respectively, and we call $f$ the integrand. Then the definite integral

is defined to be the area under the graph of $f$ between $x=a$ and $x=b$. This area is defined in the process below to be the limit of the total area of lower (inscribed) rectangles as the partition of $[a,b]$ is refined. Areas below the $x$-axis are counted negative. If $f(x)\geq 0$ for all $x$, then $\int_{a}^{b}f(x)\,dx\geq 0.$