Use this Simpson’s Rule calculator tool to accurately estimate the area under a curve using a numerical approximation method.

## How to Use the Simpson’s Rule Calculator

To use this calculator, simply fill in the function *f(x)* that you wish to integrate over a specified interval. The function should be in terms of x and can include standard mathematical operations and powers (e.g., 3*x^2 for ( 3x^2 )). Input the start and end values of the interval, and the number of subintervals (*n*) which should be an even number. Then, click the “Calculate” button to get the result.

## How the Calculator Works

The calculator uses Simpson’s Rule for numerical integration to approximate the definite integral of a function over an interval from *a* to *b*. The function input by the user is integrated using the following formula:

( int_{a}^{b} f(x) dx approx frac{h}{3} [f(a) + 4 sum f(x_{2i}) + 2 sum f(x_{2i-1}) + f(b)] )

Where *h* is the width of each subinterval ((b-a)/n), and *n* is the number of subintervals. The calculator multiplies the function values at certain points by 4 or 2, based on their position in the sequence (odd or even), then sums them up before multiplying by *h/3* for the final result.

## Limitations of Simpson’s Rule Calculator

This calculator is designed to give an approximate result. The accuracy of the approximation depends largely on the function being integrated and the number of subintervals used. As a general rule, the more subintervals, the more accurate the result, however, this also increases computational effort. Furthermore, Simpson’s Rule assumes the function is smooth over the interval; significant discontinuities or sharp turns may lead to less accurate results.