Simpson's Rule Calculator
Approximate definite integrals with Simpson's 1/3 rule and compare with the trapezoid rule.
Enter your values
Results update instantly as you type — no submit needed.
Results
- Simpson's rule estimate
- 3.24123804
- Δx (h)
- 0.25
- Subintervals used
- 8
- Trapezoid rule estimate
- 3.25174401
- Approximate error
- 0.00007122
Compared with a high-resolution Simpson run.
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.
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.