The definite integral of f(x) is the difference between two values of the integral of f(x) for two distinct values of the variable x. If the integral of f(x) dx = F(x) + C, the definite integral is denoted by the symbol

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

The quantity F(b) - F(a) is called the definite integral of f(x) between the limits a and b or simply the definite integral from a to b. It is called the definite integral because the result involves neither x nor the constant C and therefore has a definite value. The numbers a and b are called the limits of integration, a being the lower limit and b the upper limit.

### General Properties of Definite Integral

- The sign of the integral changes if the limits are interchanged.

$\displaystyle \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$ - The interval of integration may be broken up into any number of subintervals, and integrate over each interval separately.

$\displaystyle \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$ - The definite integral of a given integrand is independent of the variable of integration. Hence, it makes no difference what letter is used for the variable of integration.

$\displaystyle \int_a^b f(x) \, dx = \int_a^b f(z) \, dz$