Definite integral.

Definition 8.3.1   Let $f$ be a function defined on the interval $[a,b]$ . Suppose that $f$ has a primitive $F$ on $[a,b]$ . The (definite) integral of $f$ over $[a,b]$ is the number

$$\int_a^b f(x) \; dx = F(b)-F(a)$$

Notation: $\int_a^b f(x) \; dx = \left[ F(x) \right]_a^b$ .

Example 8.3.2       

• $\int_0^1 x^2 \; dx = \left[ \frac 13 x^3 \right] _0^1 = \frac 13 (1^3 -0^3 ) = \frac 13$ .
• $\int_0^{\pi / 2} \cos 2x \; dx = \left[ \frac 12 \sin 2x \right] = \frac 12 (\sin \pi - \sin 0 ) =0$ .