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Riemann Sum Formula
In the Riemann sum formula, we find an approximation of a region's area under a curve on a graph, commonly known as integral. Riemann's sum introduces a precise definition of the integral as the limit of a series that is infinite. Approximating the region's area of lines or functions on a graph is a very commonly used application of the Riemann sum formula. Riemann's sum formula is also used for curves. The idea of calculating the sum is obtained by adding up the areas of multiple simplified slices of the region, the general shapes that are used as multiple simplified slices of the region are rectangle, squares, parabolas, cubics, etc. Let us learn about the Riemann sum formula and a few solved examples in the upcoming sections.
What is Riemann Sum Formula?
A few methods that are used for finding the area in the Riemann sum formula:
 Right and Left methods: is used to find the area using the endpoints of left and right of the subintervals, respectively.
 Maximum and minimum methods: With this method, the values of the largest and smallest endpoint of each subinterval can be calculated.
The Riemann Sum formula is:
Sn=\(\sum_{n}^{i1}\int (x_{i})(x_{i}x_{i1})\)
Where,
 [a,b] = Closed interval divided into ‘n’ sub intervals
 f(x) = continuous function on interval
 x_{i} = Point belonging to the interval [a,b]
 f(x_{i}) = Value of the function at at x = x_{i}
Let's have a look at solved examples to understand the Riemann sum formula better
Solved Examples Using Riemann Sum Formula

Example 1: Find the area under the curve f(x) = x^{2}, 1 ≤ x ≤ 4 by using riemann sum formula with 3 intervals.
Solution:
Given : a=1
b=4
n=3
f(x) = x^{2}\(\Delta = \dfrac{ba}{n} = \dfrac{41}{3} =1 \)
By Riemann sum formula.
Sn=\(\sum_{n}^{i1}\int (x_{i})(x_{i}x_{i1})\)Area = \(\sum_i f(x_i) \Delta{x}\) = ∆x(f(2) + f(3) + f(4))
(1)(4 + 9 + 16) = 29Answer: The area under the curve is 29.

Example 2 : Find the area under the curve f(x) = x^{3}, 4 ≤ x ≤ 14 by using riemann sum formula with 5 intervals.
Solution:
Given : a=4
b=14
n=5
f(x) = x^{3}
\(\Delta = \dfrac{ba}{n} = \dfrac{144}{5} =2 \)By Using Riemann sum formula.
Sn=\(\sum_{n}^{i1}\int (x_{i})(x_{i}x_{i1})\)
Area = \(\sum_i f(x_i) \Delta{x}\)
= ∆x(f(6) + f(8) + f(10) + f(12) + f(14))
(2)(216 + 1000 + 1728 + 2744)
= 6436Answer: The area under the curve is 6436.
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