Simpsons Rule Calculator

Using the Simpson’s Rule Calculator: A Step-by-Step Guide

This guide will walk you through the steps of using the Simpson’s Rule Calculator to estimate the integral of a function over a specified interval. Follow the instructions below to ensure accurate results.

Step 1: Input Parameters

Begin by providing the necessary input parameters to the calculator. These include the lower limit, upper limit, number of intervals, and the function to be integrated. Detailed instructions are provided below:

• Lower Limit (a): Enter a numerical value representing the lower bound of the interval. This value is required.
• Upper Limit (b): Enter a numerical value representing the upper bound of the interval. This value is required.
• Number of Intervals (n): Provide an even number for the intervals. This number is critical as Simpson’s Rule requires it to be even, with a minimum accepted value of 2.
• Select Function f(x): Choose a function from the available options. The options include x², sin(x), cos(x), e^x, 1/x, and √x. Only one function can be selected, and this choice is required for the calculation.

Step 2: Understanding the Results

Once you have provided the input values, the calculator will automatically compute the required results, which include the Step Size, Integral Result using Simpson’s Rule, and an Error Estimate. The explanations of these outputs are provided below:

• Step Size (h): This value is calculated using the formula (h = frac{b-a}{n}). It indicates the width of each subinterval used in the integration process. The step size will be displayed with six decimal places for precision.
• Integral Result: This is the estimated value of the integral using Simpson’s Rule. The calculator applies the rule’s formula, which involves evaluating the function at different points and aggregating the values accordingly. The result is shown with six decimal places.
• Error Estimate: Although Simpson’s Rule provides an approximation, it also includes an error estimate calculated with the formula (|frac{h^4 cdot (b-a)}{180}|). This estimate helps you understand the potential deviation in the approximation, and it is displayed with eight decimal places.

Step 3: Review and Interpret Results

Finally, review all the calculated results for accuracy and use these to interpret or make further decisions based on your specific problem or analysis requirements. The detailed output provides a solid approximation of the definite integral, allowing you to proceed with confidence in your calculations.

With these steps, you can effectively leverage the Simpson’s Rule Calculator to aid in your numerical integration tasks. Ensure all input values are correctly provided to achieve the best possible results from the calculator.