Properties of Riemann Integral

Riemann integral possesses several important properties that make it useful in calculus and analysis. Here are some key properties:

Linearity

The Riemann integral is linear, meaning that it satisfies the properties of additivity and scalar multiplication. That is, for functions f(x) and g(x) and constants c and d, we have:

[Tex] \int_{a}^{b} (c f(x) + d g(x)) \, dx = c \int_{a}^{b} f(x) \, dx + d \int_{a}^{b} g(x) \, dx [/Tex]

Additivity

The integral of a sum of functions is the sum of their integrals. That is, for functions f(x) and g(x), we have:

[Tex]\int_{a}^{b} (f(x) + g(x)) \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx [/Tex]

Monotonicity

Monotonicity property of the Riemann integral sates that if one function is always greater than or equal to another function over an interval, then the integral of the first function should be greater than or equal to the integral of the second function over that interval. Let f and g be two Riemann integrable functions on the closed interval [a, b]. If f(x) ≤ g(x) for all x ∈ [a, b], then:

[Tex] \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx[/Tex]

Constant Multiple Rule

The integral of a constant multiple of a function is the constant multiplied by the integral of the function. That is, for a constant c, we have:

[Tex] \int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx[/Tex]

Interval Splitting

The integral over a sum of intervals is the sum of the integrals over each individual interval. That is, for intervals [a,c] and [c,b], we have:

[Tex]\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx[/Tex]

Order Preservation

If f(x) is non-negative on an interval [a,b] and f(x) ≥ g(x) for all x in [a,b], then:

[Tex]\int_{a}^{b} f(x) \, dx \geq \int_{a}^{b} g(x) \, dx[/Tex]

Riemann integral is a method used in calculus to find the area under a curve, or the total accumulated quantity represented by the curve, between two specified points. In this article, we will understand the Riemann sums, formula of Riemann integral, properties of Riemann integral and applications of Riemann integral. At the end of this article, we have given solved and practice questions for better understanding.