Simpson's Rule Calculator
Approximate ∫ f(x) dx with Simpson's 1/3 rule and see the full table of points and weights.
Updated · How we check our maths
- Simpson's rule estimate
- 5.86827358
- Step size h
- 0.523599
- Trapezoid rule (same n)
- 5.64436367
- Difference
- 0.22390991
- x0 = 0 · weight 1
- f = 0
- x1 = 0.5236 · weight 4
- f = 0.137078
- x2 = 1.0472 · weight 2
- f = 0.949703
- x3 = 1.5708 · weight 4
- f = 2.467401
- x4 = 2.0944 · weight 2
- f = 3.798813
- x5 = 2.618 · weight 4
- f = 3.426946
- x6 = 3.1416 · weight 1
- f = 0
Quick answer
Simpson's rule estimates a definite integral by fitting parabolas through groups of three points: ∫ₐᵇ f(x) dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(xₙ)], where h = (b − a)/n and n is even. For x²·sin(x) from 0 to π with n = 6 it gives about 5.8683; the exact answer is π² − 4 ≈ 5.8696.
What the Simpson's Rule Calculator does
Enter any function of x, the limits and an even number of subintervals. The calculator lists every point x₀…xₙ with its Simpson weight (1, 4, 2, 4 … 4, 1), adds them up and multiplies by h/3. It also shows the trapezoid-rule result with the same points, so you can see how much the parabola fit improves the estimate.
Simpson's rule is exact for polynomials up to degree 3, and its error shrinks with h⁴ — doubling n usually makes the error about 16 times smaller.
The formula
S = (h/3) · [f(x₀) + 4Σf(x_odd) + 2Σf(x_even, interior) + f(xₙ)], h = (b − a)/n, n even.
Error bound: |E| ≤ (b − a)·h⁴·max|f⁽⁴⁾(x)| / 180.
Worked example
∫₀² x³ dx with n = 2: h = 1, points 0, 1, 2 → f = 0, 1, 8. S = (1/3)(0 + 4·1 + 8) = 4. The exact value is 2⁴/4 = 4 — Simpson's rule is exact for cubics.
How to use the Simpson's Rule Calculator
- Type the function using x (for example
exp(-x^2)). - Enter the limits a and b — numbers, pi or e are fine.
- Choose an even n; larger n gives a more accurate answer.
- Read the estimate, and compare it with the trapezoid rule result.
Frequently asked questions
Why must n be even?
Simpson's 1/3 rule fits one parabola across every pair of subintervals, so the subintervals must come in pairs.
How accurate is Simpson's rule?
Very accurate for smooth functions: the error falls with h⁴. It is exact for any polynomial of degree 3 or less.
What is the difference between Simpson's 1/3 and 3/8 rule?
The 3/8 rule fits cubics through groups of four points and needs n to be a multiple of 3. Both have similar accuracy; the 1/3 rule is the common default.
Does it work for improper integrals?
Not directly — f(x) must be finite at every sample point. Split the interval or use a substitution first.
Results are estimates for planning and learning. Check important figures with a qualified professional. How we build and check our calculators.