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
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,
\[\int_{a}^{b} (f + g)\] = \[\int_{a}^{b} f\] + \[\int_{a}^{b} g\]
Monotonicity
If a condition is satisfied such that f<= g then,
\[\int_{a}^{b} f\] ≤ \[\int_{a}^{b} g\] is also satisfied.
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
FAQs on Riemann Integral
1. What is the fundamental concept of a Riemann Integral?
The Riemann Integral is a method in calculus for defining the exact area under a curve. It works by approximating this area with a series of rectangles. The process involves partitioning the interval into smaller subintervals, calculating the sum of the areas of these rectangles (known as a Riemann sum), and then taking the limit as the width of the subintervals approaches zero. If this limit exists and is the same regardless of how the rectangles are chosen, the function is said to be Riemann integrable.
2. How is a Riemann Integral actually calculated using partitions and sums?
The calculation of a Riemann Integral involves these key steps:
- Partition the Interval: The interval [a, b] over which you are integrating is divided into 'n' smaller subintervals. This is called a partition.
- Form Riemann Sums: For each subinterval, a rectangle is constructed. The sum of the areas of these rectangles is the Riemann sum. This involves calculating the Lower Riemann Sum (using the minimum value of the function in each subinterval) and the Upper Riemann Sum (using the maximum value).
- Refine the Partition: The process is repeated by making the partition finer, meaning the width of the largest subinterval gets smaller and smaller.
- Take the Limit: A function is integrable if, as the partition gets infinitely fine, the Lower and Upper Riemann Sums converge to the same unique value. This common value is the Riemann Integral.
3. What is the key theorem or condition that determines if a function is Riemann integrable?
The primary theorem for Riemann integrability states that a function defined on a closed and bounded interval [a, b] is Riemann integrable if and only if it is bounded and the set of its points of discontinuity has a measure of zero. A simpler, very common condition that guarantees integrability is if the function is continuous on the closed, bounded interval [a, b]. All continuous functions on such an interval are Riemann integrable.
4. What are some important properties of the Riemann Integral?
The Riemann Integral has several key properties that are useful in calculations, including:
- Linearity: The integral of a sum of functions is the sum of their integrals. Also, the integral of a constant times a function is the constant times the integral of the function.
- Additivity: If you integrate a function from 'a' to 'c' and then from 'c' to 'b', the sum is equal to the integral from 'a' to 'b'.
- Monotonicity: If f(x) ≤ g(x) on an interval [a, b], then the integral of f(x) is less than or equal to the integral of g(x) over that same interval.
- Zero Width: The integral of any function over an interval [a, a] is zero.
5. How is a Riemann Integral different from the definite integral taught in high school?
While they often yield the same result, they are conceptually different. The definite integral taught in high school is typically defined using the Fundamental Theorem of Calculus, which connects integration to differentiation (finding an antiderivative). The Riemann Integral, on the other hand, is defined from first principles using the concept of approximating area with the limit of Riemann sums (partitions and rectangles). The Riemann Integral provides the rigorous mathematical foundation for why the Fundamental Theorem works for continuous functions.
6. What are some practical applications where the concept of Riemann integration is used?
The Riemann Integral is a foundational concept with wide-ranging applications in maths, science, and engineering. Some key examples include:
- Physics: Calculating the total distance travelled by an object by integrating its velocity over time, or finding the work done by a variable force.
- Engineering: Determining the volume of irregularly shaped objects, calculating the centre of mass, or finding the pressure exerted by a fluid.
- Economics: Computing consumer surplus and producer surplus by finding the area between supply and demand curves.
- Probability and Statistics: Defining the expected value of a continuous random variable.
7. Why would a function fail to be Riemann integrable? Can you give an example?
A function can fail to be Riemann integrable for two primary reasons: it is unbounded on the interval, or it has too many discontinuities. For instance, the function f(x) = 1/x is not Riemann integrable on the interval [0, 1] because it is unbounded as x approaches 0. A classic example of a bounded function that is not Riemann integrable is the Dirichlet function, which is 1 for rational numbers and 0 for irrational numbers. Its value oscillates so erratically that the upper and lower Riemann sums never converge to the same value.
8. What is the main difference between a Riemann Integral and a Lebesgue Integral?
The core difference lies in how they partition the function's domain and range. The Riemann Integral partitions the x-axis (domain) into subintervals and approximates the area using the height of the function within those intervals. In contrast, the Lebesgue Integral partitions the y-axis (range) and then measures the 'size' (using Lebesgue measure) of the corresponding sets on the x-axis. This makes the Lebesgue Integral more powerful and able to integrate a wider class of functions, including some that are not Riemann integrable, like the Dirichlet function.
