The Importance of Subintervals in Approximating Integrals

In the field of numerical analysis, the composite trapezoid rule is a widely used technique for approximating definite integrals. This article focuses on determining the number of subintervals needed to approximate the integral of sin(x)/x over the interval [0, 1] with an error not exceeding 0.00005. We will delve into the process and provide a detailed explanation of the steps involved.

Understanding the Composite Trapezoid Rule

The composite trapezoid rule approximates the definite integral of a function by dividing the interval into subintervals and applying the trapezoid rule to each. The error in this approximation can be bounded using the error formula:

E ≤ frac{b-a^3}{12n^2}M

Where n is the number of subintervals, a and b are the bounds of the interval, and M is the maximum value of the second derivative of the function on the interval.

Step 1: Calculating the Second Derivative

To apply the error formula, we first need to find the second derivative of our function, sin(x)/x.

First Derivative: Using the quotient rule, we get: fx (x cos x - sin x)/x^2 Second Derivative: Differentiating fx again, we obtain a more complex expression involving trigonometric and standard polynomial functions.

For practical purposes, we can approximate M by evaluating the second derivative numerically or estimating its maximum value in the interval [0, 1]. In this case, M can be estimated to be around 1, but for accurate results, it is recommended to use numerical methods to find the exact value.

Step 2: Applying the Error Formula

Assuming M ≈ 1, the error formula becomes:

E ≤ frac{1-0^3}{12n^2} frac{1}{12n^2}

We want this error to be less than or equal to 0.00005:

(frac{1}{12n^2} ≤ 0.00005)

Step 3: Solving for n

Rearranging the inequality:

12n^2 ≥ frac{1}{0.00005} 20000

Dividing by 12:

n^2 ≥ frac{20000}{12} ≈ 1666.67

Taking the square root:

n ≥ sqrt{1666.67} ≈ 40.82

Since n must be an integer, we round up:

n ≥ 41

Conclusion

To approximate the integral of sin(x)/x on the interval [0, 1] using the composite trapezoid rule with an error not exceeding 0.00005, you will need at least 41 subintervals.

More on Numerical Integration Techniques

Understanding the accuracy and efficiency of numerical integration methods, such as the composite trapezoid rule, is crucial for solving complex mathematical problems in various fields, including engineering, physics, and computational science. By refining our understanding of these techniques, we can achieve more precise and reliable results in our analyses.