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Simpson's Rule Calculator N

Reviewed by Calculator Editorial Team

Simpson's Rule is a numerical method for approximating definite integrals. This calculator computes the integral of a function using Simpson's Rule with a specified number of intervals (n). It's particularly useful when exact integration is difficult or impossible.

What is Simpson's Rule?

Simpson's Rule is a numerical integration technique that approximates the area under a curve (definite integral) by fitting parabolas to segments of the curve. It's more accurate than the trapezoidal rule and requires an even number of intervals (n must be even).

The method works by:

  1. Dividing the interval [a, b] into n equal subintervals
  2. Approximating the function with parabolas on each subinterval
  3. Summing the areas of these parabolas to estimate the total area under the curve

Simpson's Rule is named after Thomas Simpson, an English mathematician who developed the method in the 18th century. It's particularly useful for integrating functions that are smooth and continuous over the interval.

How to Use the Calculator

To use the Simpson's Rule Calculator:

  1. Enter the lower limit (a) of the integral
  2. Enter the upper limit (b) of the integral
  3. Enter the number of intervals (n) - must be even
  4. Enter the function you want to integrate (e.g., x^2, sin(x), etc.)
  5. Click "Calculate" to compute the integral approximation

The calculator will display the result and optionally show a chart of the function and its approximation.

Simpson's Rule Formula

The formula for Simpson's Rule is:

∫[a,b] f(x) dx ≈ (Δx/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f(xₙ₋₂) + 4f(xₙ₋₁) + f(xₙ)]

Where:

  • Δx = (b - a)/n is the width of each subinterval
  • x₀ = a, x₁ = a + Δx, ..., xₙ = b are the endpoints of the subintervals
  • n must be an even number

For best results, the function should be smooth and continuous over the interval [a, b]. The accuracy improves as n increases, but computational cost also increases.

Worked Example

Let's calculate the integral of f(x) = x² from 0 to 2 using Simpson's Rule with n = 4 intervals.

  1. Calculate Δx = (2 - 0)/4 = 0.5
  2. Evaluate the function at the points:
    • x₀ = 0 → f(0) = 0
    • x₁ = 0.5 → f(0.5) = 0.25
    • x₂ = 1 → f(1) = 1
    • x₃ = 1.5 → f(1.5) = 2.25
    • x₄ = 2 → f(2) = 4
  3. Apply Simpson's Rule formula:
    (0.5/3) [0 + 4(0.25) + 2(1) + 4(2.25) + 4] = (0.1667) [0 + 1 + 2 + 9 + 4] = 2.333

The exact value of ∫[0,2] x² dx is 2.666..., so our approximation of 2.333 is reasonable for n=4.

FAQ

What is the difference between Simpson's Rule and the Trapezoidal Rule?

Both methods approximate integrals numerically, but Simpson's Rule uses parabolas (quadratic functions) to approximate the curve, while the Trapezoidal Rule uses straight lines. Simpson's Rule is generally more accurate for smooth functions.

Why must n be even in Simpson's Rule?

The method requires an even number of intervals to properly pair the function evaluations in the formula. Each parabola segment needs two points to define its shape, which requires an even number of intervals.

When should I use Simpson's Rule instead of exact integration?

Use Simpson's Rule when the function is complex or when an exact antiderivative is difficult or impossible to find. It's particularly useful for numerical approximations in engineering, physics, and other applied sciences.