Boole’s rule is a numerical method used to approximate the value of a definite integral. It’s a type of Newton-Cotes formula, which approximates the integral of a function by interpolating the values of the function at equally spaced points.

Boole’s rule is particularly useful when dealing with integrands that have rapidly varying curvature or oscillations.

The formula for Boole’s rule is as follows:

∫ab​f(x)dx≈452h​[7f(a)+32f(a+h)+12f(a+2h)+32f(a+3h)+7f(b)]

where:

• $a$ and $b$ are the lower and upper limits of integration, respectively.
• h=4b−a​ is the step size (assuming the integral is divided into four equal intervals).
• $f(x)$ is the integrand function.

In Boole’s rule, the interval [a,b] is divided into four subintervals, and the function values are evaluated at five equally spaced points within each subinterval. The coefficients in the formula are predetermined weights that result in an accurate approximation for polynomials up to degree 4.

Here’s a brief overview of the steps to use Boole’s rule:

1. Divide the interval of integration

   $$

   [a,b] into four equal subintervals, or choose a suitable number of subintervals.

2. Calculate the step size

   $$

   h=4b−a​.

3. Evaluate the integrand function at equally spaced points within each subinterval:

   $$

   f(a),f(a+h),f(a+2h),f(a+3h),f(b).

4. Apply Boole’s rule formula to approximate the integral using the evaluated function values.
5. Sum up the results from each subinterval to obtain the final approximation for the integral.

Boole’s rule provides a relatively accurate approximation for integrals, especially for functions that exhibit oscillatory behavior or rapidly changing curvature. However, it’s important to note that like other numerical integration methods, the accuracy of Boole’s rule depends on the smoothness of the integrand function and the number of intervals used.