Definite Integral | Research & Encyclopedia Articles

Definite Integral

The definite integral of a function f on the interval from a to b, where a < b, is a unique number I that is greater than or equal to all the lower sums and less than or equal to all of the upper sums of that function. The lower and upper sums of a function f for a partition P of [a, b] are defined as L_f(P) = m₁x₁ + m₂x₂ + ... m_nx_n and U_f(P) = M₁x₁ + M₂x₂ + ... M_nx_n, where for any integer within a subinterval the ms and Ms are the minimum and maximum values of f on that particular subinterval. A definite integral is usually written as ^b_af(x)dx. The is called the integral sign, the numbers a and b are referred to as the limits of integration, and the function f appearing in the integral is called the integrand. The main difference between a definite integral and an indefinite integral is that a definite integral is defined on a definite interval, a and b, and is equal to a unique number whereas an indefinite integral is not. The definite integral is a number depending only on f, a and b. The variable x is a dummy variable in that it may be replaced by any other variable not already in use.

The definite integral of a function f is equal to the area bounded by f on one side, the x-axis, on the left by the line x = a and on the right by the line x = b. This region bounded by this space is called the region between the graph of f and the x-axis on [a, b]. Although in some situations it is possible to calculate the definite integral of f on the interval [a, b] by calculating formulas for the lower and upper sums some functions, such as x² and cosx is tedious at least and sometimes very difficult.

Although we defined the definite integral of the function f on an interval where a < b we can relate the situations where the limits of integration are reversed, that is ^a_bf(x)dx. This definite integral is related to the one we spoke about above as ^a_bf(x)dx = -^b_af(x)dx. We should also mention that the area of a line segment is equal to 0. That is ^a_af(x)dx = 0.

The upper and lower sums are also known as Riemann sums of f on [a, b]. An important attribute of a Riemann sum is that it approximates the definite integral of a function. Georg Bernhard Riemann, for whom the sums are named, was the German mathematician who in the 19th century clarified the concept of the integral while employing such sums. The first formal definition of the integral is attributed to him.