Estimating the Definite Integral Using Left-Endpoint Approximation: A Step-by-Step Guide

Left-Endpoint Approximation

Left-Endpoint Approximation is a numerical method used to estimate the value of a definite integral

Left-Endpoint Approximation is a numerical method used to estimate the value of a definite integral. It is one of the simplest and most basic techniques in numerical integration.

To understand this method, let’s start by considering a function f(x) that is continuous on the interval [a, b]. We want to find the area under the curve of f(x) on this interval.

The left-endpoint approximation divides the interval [a, b] into n subintervals of equal width, where the width of each subinterval is given by h = (b-a) / n. It then approximates the area under the curve on each subinterval by using the value of the function at the left endpoint of the subinterval.

The formula for the left-endpoint approximation is as follows:

Approximation = h * [f(a) + f(a + h) + f(a + 2h) + … + f(a + (n-1)h)]

In this formula, f(a) represents the value of the function f(x) at the left endpoint of the interval, f(a + h) represents the value of the function at the left endpoint of the second subinterval, and so on.

To compute the approximate value of the definite integral using the left-endpoint approximation, we need to evaluate the function at these left endpoints and sum up their values multiplied by the width of each subinterval.

Here’s a step-by-step example to demonstrate the left-endpoint approximation:

1. Consider the function f(x) = x^2 on the interval [1, 3]. We want to estimate the definite integral of f(x) on this interval.

2. Choose the number of subintervals, n. Let’s say we choose n = 4 for this example.

3. Compute the width of each subinterval, h = (b-a) / n = (3-1) / 4 = 0.5

4. Evaluate the function at the left endpoints of each subinterval:
For the first subinterval, x = a = 1, so f(a) = f(1) = 1^2 = 1
For the second subinterval, x = a + h = 1 + 0.5 = 1.5, so f(a + h) = f(1.5) = (1.5)^2 = 2.25
For the third subinterval, x = a + 2h = 1 + 2(0.5) = 2, so f(a + 2h) = f(2) = 2^2 = 4
For the fourth subinterval, x = a + 3h = 1 + 3(0.5) = 2.5, so f(a + 3h) = f(2.5) = (2.5)^2 = 6.25

5. Calculate the left-endpoint approximation by summing the values of the function at each left endpoint multiplied by the width of each subinterval:
Approximation = h * [f(a) + f(a + h) + f(a + 2h) + f(a + 3h)]
= 0.5 * [1 + 2.25 + 4 + 6.25]
= 0.5 * 13.5
= 6.75

Therefore, the left-endpoint approximation for the definite integral of f(x) = x^2 on the interval [1, 3] is approximately equal to 6.75.

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