The Definite Integral

(See Summation and Product Notation)

Let be continuous on the interval . Partition into separate subintervals of equal length . Label the endpoints of the subintervals for (so that the th interval is ) and in each interval, select point . The definite interval of from to is

(See Limits)

1. Equal bounds $$\int_a^a f(x)d(x) = 0$$
2. Sum/difference rule $$\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$$
3. Constant multiple rule: $$\int_a^b kf(x)dx = k \int_a^b f(x)dx$$
4. Switching of bounds: $$\int_a^b f(x)dx = -\int_b^a f(x)dx$$
5. Intermediate separation rule: $$\int a_b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$$
   Note that does not have to be in
6. Integration of inequalities: $$\text{If } f(x) \ge g(x) \text{ for } x \in [a, b], \int_a^b f(x)dx \ge \int_a^b g(x)dx$$