Examples Using Riemann Sum Formula

Example 1: Choose which type of the Riemann integral is shown below in the figure. 

1. Left-Riemann Sum 
2. Right-Riemann Sum
3. Mid-point Riemann Sum 

Solution: 

Since the values of the intervals are decided according to the left-end point of the interval. This is a left-Riemann Sum 

Answer-(1).

Example 2: Calculate the Left-Riemann Sum for the function given in the figure above. 

Solution: 

Dividing the interval into four equal parts that is n = 4. The width of each interval will be, 

x₀ = 0, x₁ = 1, x₂ = 2, x₃ = 0 and x₄ = 0

The value of the function in each interval will be the value of the function at the right-end of the interval. 

⇒A = 

⇒A = 

⇒A = f(1)(2) + f(2)(2)+ f(3)(2) + f(4)(2)

⇒A = (f(1) + f(2) + f(3)+ f(4))(2)

⇒A = (1 + 2 + 3+ 4)(2)

⇒A = (10)(2)

⇒A = 20

Example 3: Consider a function f(x) = 5 – 2x, its area is calculated from riemann sum from x = 0 to x = 4, the whole area is divided into 4 rectangles. Find the riemann sum in sigma notation

Solution: 

Step (i): Calculate the width

Whole length is divided into 4 equal parts, 

x_i = 0 and x_l = 4, 

Width of an interval is given by = 

Where x_i = initial point, and x_l – last point and n= number of parts 

n = 4 

Step(ii): 

a = 0, 

x_i = 0 +  i 

⇒ x_i = i

Step (iii): 

A_i = Height x Width 

    = f(x_i) 

    = (5 – 2i)(1)

    = 5 – 2i

Total Area = A(1) + A(2) + A(3) + A(4) + A(5)

= 

Example 4: Consider a function f(x) = √x, its area is calculated from riemann sum from x = 0 to x = 4, the whole area is divided into 4 rectangles. Find the riemann sum in sigma notation

Solution: 

Step (i): Calculate the width

Whole length is divided into 4 equal parts, 

x_i = 0 and x_l = 4, 

Width of an interval is given by = 

Where x_i = initial point, and x_l – last point and n= number of parts 

n = 4 

Step(ii): 

a = 0, 

x_i = 0 + i 

⇒ x_i = i

Step (iii) 

A_i = Height x Width 

    = f(x_i) 

    = (√i)(1)

    = √i

Total Area = A(1) + A(2) + A(3) + A(4) 

= 

Example 5: Consider a function f(x) = 3(x + 3), its area is calculated from Riemann sum from x = 0 to x = 6, and the whole area is divided into 6 rectangles. Find the Riemann sum in sigma notation

Solution: 

Step (i): Calculate the width

Whole length is divided into 4 equal parts, 

x_i = 0 and x_l = 6, 

Width of an interval is given by = 

Where x_i = initial point, and x_l – last point and n= number of parts 

n = 6 

Step(ii): 

a = 0, 

x_i = 0 +  i 

⇒ x_i = i

Step (iii):

A_i = Height x Width 

    = f(x_i) 

    = (3(i + 3))(1)

    = 3(i + 3)

Total Area = A(1) + A(2) + A(3) + A(4) 

= 

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.