How to use Reimann Sums In Mathematics

Reimann Sums is calculated by dividing a given function’s graph into smaller rectangles and summing the areas of each rectangle. The more rectangles we consider by subdividing the provided interval, the more precise the area computed by this approach is; nevertheless, the more subintervals we consider, the more difficult the calculations get.

Reimann Sum can be classified into three more categories such as:

• Left Reimann Sum
• Right Reimann Sum
• Midpoint Reimann Sum

Area using the Reimann sum is given as follows:

[Tex]\bold{Area = \sum_{i=1}^{n}f(x_i)\Delta x_i}[/Tex]

where,

• f(x_i) is the value of the function being integrated at the i^th sample point
• Δx = (b-a)/n is the width of each subinterval, 
  • a and b are the limits of integration and 
  • n is the number of subintervals
• ∑ represents the sum of all the terms from i=1 to n, 

Example: Find the area under the curve for function, f(x) = x² between the limits x = 0 and x = 2.

Solution:

We want to find the area under the curve of this function between x = 0 and x = 2. We will use a left Reimann Sum with n = 4 subintervals to approximate the area.

Let’s calculate the area under the curve using 4 subintervals. 

Thus, width of subintervals, Δx = (2-0)/4 = 0.5

All the 4 subintervals are,

a = 0 = x₀ < x₁ <x₂ < x₃ < x₄ = 2 = b

x₀ = 0, x₁ = 0.5, x₂ = 1, x₃ = 1.5, x₄ = 2

Now we can evaluate the function at these x-values to find the heights of each rectangle:

f(x₀) = (0)² = 0
f(x₁) = (0.5)² = 0.25
f(x₂) = (1)² = 1
f(x₃) = (1.5)² = 2.25
f(x₄) = (2)² = 4

Area under the curve can now be approximated by summing the areas of the rectangles formed by these heights:

A ≈ Δx[f(x₀) + f(x₁) + f(x₂) + f(x₃)] = 0.5[0 + 0.25 + 1 + 2.25] = 1.25

Therefore, area under the curve of f(x) = x² between x = 0 and x = 2, approximated using a left Reimann Sum with 4 subintervals, is approximately 1.25.

Area Under Curve is area enclosed by curve and the coordinate axes, it is calculated by taking very small rectangles and then taking their sum if we take infinitely small rectangles then their sum is calculated by taking the limit of the function so formed.

For a given function f(x) defined in the interval [a, b], the area (A) under the curve of f(x) from ‘a’ to ‘b’ is given by A = ∫_a ^b f(x)dx. The area under a curve is computed by taking the absolute value of the function over the interval [a, b], summed over the range.

In this article, we will learn about, the area under the curve, its applications, examples, and others in detail.