2. Introduction to Integrals

Integration is the opposite reaction of differentiation. There is including a signed area, compounding a process, center of mass/average.

Notation

The integral of the function $f(x)$ over the interval $[a,b]$ is denoted:

${\int }_a^{b}f(x)dx$

In the notation above, $[a,b]$ is the interval of integration, $\int$ is the integral sign, $a$ is the lower bound, $b$ is the upper bound, and $f(x)$ is the integrand.

Area

Definition/Theorem

Suppose $f(x)$ is a function that is non-negative for all $x\in [a,b]$, then ${\int }_a^{b}f(x)dx=A$, where $A$ is the surface area between the graph of $f$ and the x-axis over the interval $[a,b]$.