What is Simpson's Rule?

Simpson's Rule is a numerical method used to approximate the definite integral of a function. It is named after the English mathematician Thomas Simpson (1710-1761). This method provides a more accurate approximation than simpler techniques like the trapezoidal rule because it uses quadratic polynomials to estimate the area under a curve.

The basic idea behind Simpson's Rule is to divide the area under a curve into several parabolic sections. By calculating the areas of these parabolic sections and summing them up, we can obtain a good approximation of the definite integral.

How to Use the Simpson's Rule Calculator

Step 1: Enter the Function

In the "Function f(x)" field, type in the mathematical function you want to integrate. Use standard mathematical notation, replacing "^" for exponents and "" for multiplication. For example, to enter x squared plus 2x plus 1, type "x^2 + 2x + 1".

Step 2: Specify the Integration Limits

Enter the lower limit of integration in the "Lower Limit (a)" field and the upper limit in the "Upper Limit (b)" field. These values define the interval over which you want to calculate the integral.

Step 3: Choose the Number of Subintervals

In the "Number of Subintervals (n)" field, enter an even number that represents how many subintervals you want to use in the approximation. A higher number of subintervals generally leads to a more accurate result, but also increases computation time.

Step 4: Calculate the Integral

Click the "Calculate" button. The calculator will process your input and display the approximate value of the integral.

Step 5: Interpret the Results

The calculator will show the approximate integral value and provide a brief explanation of how it was calculated, including the number of subintervals used and the integration interval.

By following these steps, you can quickly and accurately approximate definite integrals using Simpson's Rule. This tool is particularly useful for complex functions or when you need a numerical approximation of an integral that's difficult to solve analytically.