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Simpson's Rule (Single Strip) Calculator

Estimate the area under a curve using a parabola.

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Area

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Overview

Simpson's Rule is a numerical integration technique that approximates the area under a curve by fitting a quadratic parabola through three evenly spaced points. It belongs to the Newton-Cotes formulas and provides a more accurate estimation than the Trapezoidal Rule for functions that are reasonably smooth.

Symbols

Variables

h = Step Size, y_0 = First Height, y_1 = Mid Height, y_2 = Last Height, A = Area

Step Size
First Height
Mid Height
Last Height
Area

Apply it well

When To Use

When to use: Apply this rule when you need to calculate the definite integral of a function using discrete data points or when the antiderivative is difficult to find. It requires three points (the start, midpoint, and end) spanning two equal sub-intervals of width 'h'.

Why it matters: This rule is a fundamental tool in engineering and physics for approximating work, fluid pressure, and centers of mass from experimental data. It strikes a balance between computational simplicity and high-order accuracy, making it a standard algorithm in scientific software.

Avoid these traps

Common Mistakes

  • Using h/2 instead of h/3.
  • Weights order.

One free problem

Practice Problem

A civil engineer measures the depth of a river cross-section at three points spaced 6 meters apart. The recorded depths are 2 meters, 5 meters, and 2 meters. Calculate the approximate area of the cross-section using Simpson's Rule.

Step Size6
First Height2
Mid Height5
Last Height2

Solve for:

Hint: Plug the values directly into the formula A ≈ (h/3)(y0 + 4y1 + y2).

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Simpson's rule
  2. Atkins' Physical Chemistry
  3. Bird, Stewart, Lightfoot (Transport Phenomena)
  4. Halliday, Resnick, Walker (Fundamentals of Physics)
  5. Incropera, DeWitt, Bergman, Lavine (Fundamentals of Heat and Mass Transfer)
  6. Chapra, Steven C., and Raymond P. Canale. Numerical Methods for Engineers. 7th ed. McGraw-Hill, 2015.
  7. Incropera, Frank P., et al. Fundamentals of Heat and Mass Transfer. 7th ed. John Wiley & Sons, 2011.
  8. Atkins, Peter, and Julio de Paula. Atkins' Physical Chemistry. 10th ed. Oxford University Press, 2014.