Definition: The Definite Integral.

If $f(x)$ is a function defined on an interval $[a, b]$, the definite integral of $f$ from $a$ to $b$ is given by

$$
\int_a^b f(x) d x=\lim {n \rightarrow \infty} \sum_{i=1}^n f\left(x_i^*\right) \Delta x
$$

provided the limit exists. If this limit exists, the function $f(x)$ is said to be integrable on $[a, b]$, or is an integrable function.

NOTE: you can use anything for the variable of integration without changing the value, so $\int_a^b f(x)dx = \int_a^b f(u)du = \int_a^b f(t)dt$.

Example 1 (warmup):

Use the graph of f(x) above to evaluate the following:

  1. $\int_0^4 f(x)$
  2. $\int_4^6 f(x)$