Summation Notation of Riemann Sum

Steps given below should be followed to find the summation notation of the Riemann Integral.

• Step 1: Find out the width of each interval. Let’s denote the width of interval with 

• Step 2: Let x_i denote the right-endpoint of the rectangle x_i = a + .i

• Step 3: Define Area of Each Rectangle

• Step 4: Sum the Areas 

Let’s say the goal is to calculate the area under the graph of the function f(x) = x³, the area will be calculated between the limits x = 0 to x = 4.

Divide the interval into four equal parts, the intervals will be [0, 1], [1, 2], [2, 3], and [3, 4]. 

Riemann sum then, can be written as follows, 

A(1) + A(2) + A(3) + A(4) = 

Let’s calculate the right sum Riemann sum. Assume x_i denotes the right endpoint of the i^th rectangle. 

So, the formula for x_i = i. Now, the value of the function at these points becomes, 

f(x_i) = (i)³

So, A(i) = (height)(width) = (i)³

Riemann sum becomes, 

 A(1) + A(2) + A(3) + A(4) = 

⇒ A(1) + A(2) + A(3) + A(4) = 

So, this way almost all the Riemann sums can be represented in a sigma notation. 

Let’s work out some problems with these concepts. 

Articles Related to Riemann Sum:

• Trapezoidal Rule
• Riemann Sums in Summation Notation
• Definite Integral as the Limit of a Riemann Sum

Riemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums.

In this article, we will look into the Riemann sums, their approximation, sum notation, and solved examples in detail.