Numerical Integration Calculator

Evaluate a definite integral numerically by the trapezoidal rule, Simpson's 1/3 rule or Simpson's 3/8 rule, from a function or a table of values, with the ordinates, weights and the error against an accurate value.

Rule	Simpson's 1/3 rule (n even)
From	A function
f(x) =	1/(1 + x^2)
From x =	0
To x =	1
Number of intervals n	6

Rule

From

f(x) =

Angles in radians.

From x =

Use pi for π.

To x =

Number of intervals n

∫ 1 / (1 + x^2) dx from 0 to 1

≈ 0.78539795

Simpson's 1/3 rule with n = 6 and h = 0.16666667 gives 0.78539795. The accurate value is 0.7853981634, so the rule's error is 0.

Approximate. Checked numerically.

Method: Simpson's 1/3 rule
Worked out with: JavaScript floating point
Check: Every step compared at several values of x; compared with an adaptive (tanh–sinh) quadrature of the same integral

h: 0.16666667
n (intervals): 6
Accurate value (adaptive quadrature): 0.7853981634
Error of the rule: −0.00000022
Trapezoidal rule: 0.78424077
Simpson's 3/8 rule: 0.78539586

Ordinates and weights

i	xᵢ	yᵢ	weight
0	0	1	1
1	0.166667	0.972973	4
2	0.333333	0.9	2
3	0.5	0.8	4
4	0.666667	0.692308	2
5	0.833333	0.590164	4
6	1	0.5	1

The curve and the ordinates used

Point at the curvePoint at the chart, or Tab to it and use ← →

f(x)yᵢ

About the Numerical Integration Calculator

Split [a, b] into n equal intervals of width h = (b − a)/n and take the ordinates y₀, y₁, …, yₙ. The trapezoidal rule joins them with straight lines: (h/2)[(y₀ + yₙ) + 2(y₁ + … + yₙ₋₁)]. Simpson's 1/3 rule fits parabolas through each pair of intervals, so n must be even: (h/3)[(y₀ + yₙ) + 4(odd ordinates) + 2(even ordinates)]. Simpson's 3/8 rule fits cubics through groups of three intervals, so n must be a multiple of 3.

For a function, the answer is compared with an accurate value from adaptive quadrature, so you can see each rule's error.

The formula

Simpson's 1/3: ∫ y dx ≈ (h/3)[(y₀ + yₙ) + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …)]

Worked example

∫ dx/(1 + x²) from 0 to 1, n = 6, h = 1/6

y₀ = 1, y₆ = 0.5; odd ordinates sum 2.3409…; even 1.5600…
(1/18)[1.5 + 4(2.34089) + 2(1.56)] ≈ 0.785398
Exact: π/4 = 0.785398163…

Questions people ask

Which rule is most accurate?

For smooth functions Simpson's rules are far more accurate than the trapezoidal rule for the same n: their error shrinks like h⁴ rather than h².

Why must n be even for Simpson's 1/3 rule?

It fits one parabola over each pair of intervals, so the intervals must pair up.