Definite Integral

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

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

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

  2. The interval of integration may be broken up into any number of sub-intervals, and integrate over each interval separately.

    $$\displaystyle \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$$

  3. 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.

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