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Simpson

Written by the Fiveable Content Team • Last updated August 2025
Written by the Fiveable Content Team • Last updated August 2025

Definition

Simpson's rule is a numerical method used to approximate the definite integral of a function. It is a type of quadrature formula that uses a parabolic interpolation to estimate the area under a curve, providing a more accurate approximation than the simpler Trapezoidal rule.

Simpson cheat sheet for homework

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5 Must Know Facts For Your Next Test

  1. Simpson's rule is a more accurate approximation method than the Trapezoidal rule, as it uses a parabolic interpolation instead of a linear one.
  2. Simpson's rule requires that the interval of integration be divided into an even number of subintervals, with the function evaluated at the endpoints and midpoints of each subinterval.
  3. The formula for Simpson's rule is $\int_a^b f(x) dx \approx \frac{b-a}{3}[f(a) + 4f(\frac{a+b}{2}) + f(b)]$, where $a$ and $b$ are the endpoints of the interval and $f(x)$ is the function being integrated.
  4. Simpson's rule has a truncation error that is proportional to the fourth power of the subinterval width, making it a more accurate approximation than the Trapezoidal rule, which has a truncation error proportional to the square of the subinterval width.
  5. Simpson's rule can be extended to multiple dimensions, known as the Composite Simpson's rule, which is used to approximate double and triple integrals.

Review Questions

  • Explain how Simpson's rule differs from the Trapezoidal rule in approximating the definite integral of a function.
    • The key difference between Simpson's rule and the Trapezoidal rule is the type of interpolation used to estimate the area under the curve. While the Trapezoidal rule uses a linear interpolation, Simpson's rule employs a parabolic interpolation. This makes Simpson's rule a more accurate approximation method, as it can better capture the curvature of the function being integrated. Specifically, Simpson's rule has a truncation error that is proportional to the fourth power of the subinterval width, compared to the Trapezoidal rule's truncation error being proportional to the square of the subinterval width.
  • Describe the requirements for applying Simpson's rule to approximate a definite integral.
    • To use Simpson's rule, the interval of integration must be divided into an even number of subintervals. The function being integrated must be evaluated at the endpoints of the interval, as well as the midpoints of each subinterval. The formula for Simpson's rule is then applied, which takes the weighted sum of the function values at these points to estimate the definite integral. Specifically, the formula is $\int_a^b f(x) dx \approx \frac{b-a}{3}[f(a) + 4f(\frac{a+b}{2}) + f(b)]$, where $a$ and $b$ are the endpoints of the interval and $f(x)$ is the function being integrated.
  • Explain how the Composite Simpson's rule can be used to approximate double and triple integrals, and discuss the advantages of this approach compared to other numerical integration methods.
    • The Composite Simpson's rule extends the one-dimensional Simpson's rule to higher dimensions, allowing for the approximation of double and triple integrals. By dividing the region of integration into a grid of rectangular or cuboid subregions, and applying the Simpson's rule formula to each subregion, the Composite Simpson's rule can provide a more accurate approximation of the integral compared to other numerical integration methods, such as the Composite Trapezoidal rule. The key advantage is that the Composite Simpson's rule utilizes a parabolic interpolation, which can better capture the curvature of the function being integrated over a two- or three-dimensional domain. This improved accuracy makes the Composite Simpson's rule a valuable tool for numerical integration in fields such as engineering, physics, and applied mathematics.
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