Simpson's Rule Calculator

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Approximate the definite integral of a function using parabolic arcs.

Numerical Integration Data

Y-values (comma separated)

Step Size (h or Δx)

Integration Results

41.3333

Approximate Integral

Calculated using Simpson's 1/3 Rule.

Number of Intervals (n)4

Average Height10.33

y=1

y=4

y=9

y=16

y=25

About this calculator

Quick Reference

Total Points5

Multiplier Pattern1, 4, 2, 4, ..., 1

What is Simpson's Rule?

Simpson's Rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it approximates the area under a curve by dividing it into an even number of intervals and fitting parabolas through sets of three consecutive points.

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Pro Tips

The number of intervals (n) must be EVEN.
Simpson's Rule is generally more accurate than the Trapezoidal Rule for the same number of points.
It provides exact results for polynomials up to degree 3.

Fun Facts

"Thomas Simpson was a self-taught mathematician who lived in the 18th century."
"This method is a special case of the Newton-Cotes formulas."
"In civil engineering, Simpson's Rule is used to calculate the volumes of earthworks and reservoirs."

Formula: ∫ f(x)dx ≈ (h/3) [y₀ + 4y₁ + 2y₂ + 4y₃ + ... + yₙ]