Simpson Rule Calculator

Simpson’s Rule Calculator Usage Guide

This guide will walk you through the process of using the Simpson’s Rule Calculator to approximate the integral of various mathematical functions.

Step 1: Select the Function

The first step is to choose the mathematical function you wish to integrate. Available functions include:

• x² – The square of x
• sin(x) – The sine of x
• cos(x) – The cosine of x
• e^x – The exponential function
• 1/x – The reciprocal of x
• √x – The square root of x

Select your desired function from the provided dropdown menu.

Step 2: Input the Limits of Integration

Next, you must input the lower and upper limits of integration:

• Lower Limit (a): Enter the starting point of the interval.
• Upper Limit (b): Enter the ending point of the interval.

Ensure both limits are numerical values with a precision up to four decimal places.

Step 3: Define the Number of Subintervals

The number of subintervals plays a crucial role in the accuracy of the Simpson’s Rule. You need to input the number of subintervals (n) which must be an even number, with a minimum of 2.

Enter a suitable value based on the precision you desire, keeping in mind that larger numbers increase computational accuracy.

Step 4: Calculate the Step Size (Δx)

Once all inputs are in place, the calculator computes the step size (Δx) using the formula:

Δx = (b – a) / n

This value indicates the width of each subinterval.

Step 5: Compute the Integral Using Simpson’s Rule

The calculator then applies Simpson’s Rule to compute the approximate integral:

Integral Result = (Δx/3) * (f(a) + f(b) + 4*sum(i=1:2:n-1, f(a + i*Δx)) + 2*sum(i=2:2:n-2, f(a + i*Δx)))

This result is presented as a numerical value with six decimal places of precision.

Step 6: Review the Estimated Accuracy

Lastly, the tool calculates the estimated accuracy of the integral approximation as a percentage:

Estimated Accuracy = |1 – (Integral Result / Actual Value)| * 100

This provides insight into the reliability of the approximation, displayed with four decimal places for precision.

By following these steps, you can effectively use the Simpson’s Rule Calculator to estimate integrals with precision and ease.