Math

Simpson's Rule Calculator

Approximate definite integrals with Simpson's 1/3 rule and compare with the trapezoid rule.

Enter your values

Function f(x)

Lower limit a

Upper limit b

Subintervals n (even)

ⓘ Live results

Results

Simpson's rule estimate: 3.24123804
Δx (h): 0.25
Subintervals used: 8
Trapezoid rule estimate: 3.25174401
Approximate error: 0.00007122

x = 0f(x) = sqrt(1 + x^3)x = 2

i	xᵢ	f(xᵢ)	weight
0	0	1	1
1	0.25	1.007782	4
2	0.5	1.06066	2
3	0.75	1.192424	4
4	1	1.414214	2
5	1.25	1.718466	4
6	1.5	2.09165	2
7	1.75	2.52178	4
8	2	3	1

Quick answer

Simpson's rule approximates a definite integral by fitting parabolas through pairs of subintervals: ∫f ≈ (Δx/3)[f₀ + 4f₁ + 2f₂ + … + 4f_{n−1} + f_n], where n is even.

∫ₐᵇ f(x) dx ≈ (Δx/3)[f₀ + 4f₁ + 2f₂ + ⋯ + 4fₙ₋₁ + fₙ]

How to use the Simpson's Rule Calculator

1Enter f(x) and the limits of integration.
2Choose an even number of subintervals — more subintervals means more accuracy.
3Read the estimate, Δx and the weighted table of sample points.

Worked example

Using the values the calculator loads with:

Inputs

Function f(x)sqrt(1 + x^3)
Lower limit a0
Upper limit b2
Subintervals n (even)8

Results

Simpson's rule estimate3.24123804
Δx (h)0.25
Subintervals used8
Trapezoid rule estimate3.25174401
Approximate error0.00007122

Frequently asked questions

Why must n be even for Simpson's rule?

Each parabola spans two subintervals, so the interval count has to be divisible by two. If you enter an odd n this calculator rounds up to the next even number.

Is Simpson's rule more accurate than the trapezoid rule?

Almost always. Simpson's rule is exact for cubics and has error on the order of h⁴, while the trapezoid rule error is on the order of h².

About this calculator

Built and reviewed by James Whitfield · How we build and test calculators
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Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.