Enter the width of the subintervals, function to be integrated, and the lower and upper limits of the integral into the calculator to determine the numerical approximation of the definite integral using Simpson's 3/8 Rule.

Simpson's 3/8 Rule Formula

The following formula is used to calculate the numerical approximation of a definite integral using Simpson's 3/8 Rule. 

I = (3h/8) * (f(a) + 3f(a+h) + 3f(a+2h) + f(b))

Variables:

  • I is the numerical approximation of the definite integral
  • h is the width of the subintervals (b-a)/n
  • f(x) is the function to be integrated
  • a and b are the lower and upper limits of the integral respectively
  • n is the number of subintervals

To calculate the numerical approximation of the definite integral using Simpson's 3/8 Rule, first calculate the width of the subintervals by subtracting the lower limit of the integral from the upper limit and dividing by the number of subintervals. Then, multiply the width of the subintervals by 3/8. Multiply this result by the sum of the function evaluated at the lower limit, three times the function evaluated at the lower limit plus the width of the subintervals, three times the function evaluated at the lower limit plus twice the width of the subintervals, and the function evaluated at the upper limit.

What is Simpson's 3/8 Rule?

Simpson's 3/8 Rule is a numerical integration technique for approximating definite integrals. It is a more accurate method than the standard Simpson's Rule as it uses cubic interpolation instead of quadratic, fitting a polynomial of degree three through four points instead of a polynomial of degree two through three points. This rule is particularly useful when the function to be integrated is known at a number of equally spaced points.

How to Calculate Simpson's 3/8 Rule?

The following steps outline how to calculate the numerical approximation of a definite integral using Simpson's 3/8 Rule.

  1. First, determine the width of the subintervals (h) using the formula h = (b - a) / n.
  2. Next, evaluate the function (f(x)) at the lower limit (a), the upper limit (b), and two additional points within each subinterval (a + h), (a + 2h), and (a + 3h).
  3. Next, calculate the numerical approximation of the definite integral using the formula I = (3h/8) * (f(a) + 3f(a+h) + 3f(a+2h) + f(b)).
  4. Finally, calculate the numerical approximation of the definite integral using the calculated values.
  5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Function (f(x)): x^2

Lower limit (a): 1

Upper limit (b): 3

Number of subintervals (n): 3