Simpson’s rule

Definition

Simpson's rule is a method for numerical integration that approximates the value of a definite integral by using quadratic polynomials. It is particularly useful for functions that are difficult to integrate analytically.

5 Must Know Facts For Your Next Test

1. Simpson's rule uses parabolic segments to approximate the area under a curve.
2. The formula for Simpson's rule over an interval $[a, b]$ with $n$ subintervals (where $n$ must be even) is given by: $$\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{3n} [f(a) + 4 \sum_{i=1,3,5,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,...}^{n-2} f(x_i) + f(b)]$$
3. Simpson's rule requires that the number of subintervals (n) be even.
4. It provides more accurate results compared to the trapezoidal rule when applied to smooth functions.
5. The error term in Simpson's Rule is proportional to the fourth derivative of the function being integrated.

Review Questions

• What condition must be satisfied regarding the number of subintervals used in Simpson's rule?
• How does Simpson’s rule compare in accuracy to the trapezoidal rule?
• Write down and explain the formula for Simpson’s rule.

Related terms

Trapezoidal Rule: A method of numerical integration that approximates the area under a curve using trapezoids.

Numerical Integration: Techniques used to approximate the value of definite integrals when they cannot be computed analytically.

Midpoint Rule: A method of numerical integration that approximates the area under a curve by evaluating function values at midpoints of subintervals.