Simpson's Rule Calculator
What is Simpson’s Rule?
Simpson's rule approximates definite integrals using parabolic arcs instead of straight lines. It offers higher accuracy than trapezoidal rule, especially for smooth functions. The composite Simpson's 1/3 rule divides interval [a,b] into an even number n of subintervals.
Formula
∫ₐᵇ f(x) dx ≈ (h/3)[f(x₀)+4∑f(x_odd)+2∑f(x_even)+f(xₙ)] where h=(b-a)/n, n even. Error bound: |E| ≤ (b-a)⁵·M/(180 n⁴) where M = max|f⁽⁴⁾(x)|.
1/3 vs 3/8 Rule
Simpson's 1/3 rule uses quadratic interpolation (3 points). Simpson's 3/8 rule uses cubic (4 points), useful when n is multiple of 3. Our calculator implements the widely-used composite 1/3 rule for even n.
How to Calculate Simpson's Rule – Step-by-Step (2026)
Example: Approximate ∫₀² x² dx using Simpson's rule with n=4. Exact value = 8/3 ≈ 2.666667.
- Step 1: h = (2-0)/4 = 0.5
- Step 2: x₀=0, x₁=0.5, x₂=1, x₃=1.5, x₄=2 → f(x)=x² → y: 0, 0.25, 1, 2.25, 4
- Step 3: Simpson's sum = (h/3)[f₀ + f₄ + 4(f₁+f₃) + 2(f₂)] = (0.5/3)[0+4 + 4(0.25+2.25) + 2(1)] = (0.16667)[4 + 4(2.5) + 2] = 0.16667*(4+10+2)=0.16667*16=2.66667 ✓
Pro tip: Always ensure your n is even for 1/3 rule. Use our calculator's error estimate to gauge accuracy.
Comparison: Simpson’s Rule vs Trapezoidal Rule
| Method | Convergence Order | Error Term | Best for |
|---|---|---|---|
| Trapezoidal Rule | O(1/n²) | −(b−a)³ f''(ξ) / (12n²) | Rough approximations |
| Simpson's 1/3 Rule | O(1/n⁴) | −(b−a)⁵ f⁽⁴⁾(ξ) / (180 n⁴) | Smooth functions, higher accuracy |
| Composite Simpson | O(1/n⁴) | Very small for moderate n | Scientific computing, AP calculus |
Simpson's Rule for Arc Length & Area Calculation
Beyond standard integration, Simpson's rule can estimate arc length using L = ∫ √(1+[f'(x)]²) dx. You can apply our calculator by defining f(x) as √(1+(dy/dx)²). For area between curves, simply integrate the difference. Many students use a simpson's rule calculator with table when experimental data points are given — our Table Mode handles exactly that.
On TI-84, you can program the formula or use summation features. On Desmos, use the calculator above as an online alternative. Our online simpson's rule calculator gives instant step-by-step for homework help in 2026.
Frequently Asked Questions
Simpson's 1/3 uses quadratic polynomials (3 points per panel) and requires even number of intervals. Simpson's 3/8 uses cubic polynomials (4 points) and multiples of 3 intervals. Our calculator focuses on 1/3 rule for general use.
Enter comma-separated X and Y values in Table Mode. Ensure points are equally spaced and the count is odd. Click Compute — the tool applies composite Simpson's rule automatically.
Yes, but it works best for smooth functions with continuous fourth derivative. For highly oscillatory functions, increase n or use adaptive methods.
We provide an empirical error estimate by comparing with double-interval computation (Romberg style). It gives a reliable magnitude for typical functions.
The composite rule pairs intervals; each parabola spans two subintervals. An odd n would leave an unpaired interval requiring a different method.
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It applies Simpson's rule over multiple subintervals to increase accuracy. Our tool uses the composite approach automatically.