Definition of Integral

Let $f$ be a continuous function on the interval $[a,b]$. The definite integral is just the signed area between the graph of $f$ and the $x$ axis:

Definition 2.1.2 (Definite Integral)   The definite integral of $f(x)$ from $a$ to $b$ is

$$\int_{a}^{b} f(x) dx = \lim_{n\to\infty}\left( \sum_{i=1}^n f(x_i^*) \Delta x \right).$$

Properties of Integration:

• $\int_{a}^b f(x) dx = - \int_{b}^{a} f(x) dx$
• $\int_{a}^b c_1 f_1(x) + c_2 f_2(x) dx = c_1 \int_{a}^b f_1(x) + c_2 \int_{a}^{b} f_2(x) dx.\qquad$ (linearity)
• If $f(x)\geq g(x)$ on for all $x\in [a,b]$, then $\int_{a}^b f(x) dx \geq \int_{a}^b g(x) dx$.

There are many other properties.