Riemann Integral

Riemann integral was actually developed by a German Mathematician Bernhard Riemann who made significant contributions to differential geometry, number theory and its analysis and rose to fame for his rigorous formulation of the Riemann integral. It was introduced for the analysis in the study of functions for real variables. Real variables, real valued functions and real numbers are what it deals with. Along with the study of analytic properties of sequences, it also includes the study of limits and the convergence of real number series and sequences. The analysis of this study deals with the properties of the real valued functions including continuity and is also related to the calculus of real numbers. 

What is Riemann Integral?

Supposing that f(x) is a continuous and a non-negative function over which has a range of [a, b]. The area between f and x-axis represents integral of “f” with respect to x. This type of integral is known as a definite integral for function f in the closed interval [a, b]. if, the function is almost constant in all the sub intervals, the Riemann Integral works in such situations. 

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According to the Riemann integral definition and its method of calculating the integral, this area can be calculated by dividing the area into a series of rectangles, then finding the areas of those rectangles and subsequently adding them to get a total final area value for the definite integral.

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If the rectangles formed are extremely narrow then evidently, the approximation error will become negligible. The value obtained from the measurement of the areas of narrow rectangles may be similar to the value of the area under the curve f. This method will break down if we make larger rectangles. Hence, to make this method more precise, the width of the rectangles can be determined by breaking the interval [a, b] into smaller parts. 

Important Riemann Sum Terms

Some terms that need to be understood before proceeding with Riemann sums. 

• Partition: Let [a, b] ∈ R be a closed interval. The partition of this interval can have the sequence of this form:  a= x0 < x1 < ………< xn where every [xi, xi+1] is called a sub interval.

• Norm: It is defined as the length of the biggest sub interval. Also called a mesh. 

• Tagged Partition: Let [a, b] ∈ R be a closed interval. Partition q(x, t) along the numbers t0, t1, ……t n-1 is called a tagged partition. It needs to satisfy a condition that for every i, ti ∈ [xi, xi+1]. 

Riemann Sum 

It is defined as the sum of real valued function f in the interval [a,b] with respect to the tagged partition of [a, b]. The formula for Riemann sum is as follows: 

\[\sum_{i=0}^{n-1} f(t_{i}) (x_{i+1} - x_{i})\]

Each term in the formula is the area of the rectangle with length/height as f(ti) and breadth as xi+1- xi. 

So, the final value of Riemann sum is the sum of areas of all the rectangles which is actually the area under the curve of f(x) within the interval [a, b].

Riemann Integral Formula

Let us suppose that we have a real valued function “f” spanning over the range [a, b] and “L” a real number. 

The function “f” is said to be eligible for integration in the interval [a, b] only under the following condition: 

There exists a δ such that δ > 0 for each ϵ > 0.  Now, for each partition which has a property ||P|| < δ we can say that: 

|S (f, P) - L| < ϵ

If L is the integral of f within the interval [a, b], we can write it as: 

L =  \[\int_{a}^{b} f(x)dx\]

Properties of Riemann Integral

Riemann integrals have three important properties which are: 1) Linearity, 2) Monotonicity and 3) Additivity. 

Let us look at each one of them in detail

1. Linearity 

If f: [a, b] → R is integrable

And there exists a c ϵ R, the we can say that cf is also integrable such that: 

\[\int_{a}^{b} cf\] =\[c\int_{a}^{b} f\] and,

 \[\int_{a}^{b} (f + g)\] = \[\int_{a}^{b} f\] + \[\int_{a}^{b} g\] 

2. Monotonicity 

If a condition is satisfied such that f<= g then, 

\[\int_{a}^{b} f\] ≤ \[\int_{a}^{b} g\] is also satisfied.

3. Additivity 

If a condition is satisfied such that a < c < b then, 

\[\int_{a}^{c} f\] + \[\int_{c}^{b} f\] = \[\int_{a}^{b} f\] is also satisfied.

Riemann Sum Example 

Example 1) Show that f is integrable using the Riemann conditions and criterion. 

Where f (x) = x on [0,1]

Solution) Let Pn = {0, 1 /n, 2/ n, ..., n−1/n, n/n} 

Then U (Pn, f) − L (Pn, f) = \[\frac{n}{n}^{2}\] − n − \[\frac{1}{n}^{2}\] → 0