Simpson's Rule Calculator – Approximate Integrals Easily
Use Simpson's Rule to approximate definite integrals with high accuracy. This numerical integration method provides excellent approximations for smooth functions using parabolic segments.
Supported: +, -, *, /, **, sin, cos, tan, sqrt, log, ln, pi, e
Simpson's 1/3 Rule Formula:
∫ab f(x)dx ≈ (b-a)/(3n) [f(x₀) + 4f(x₁) + 2f(x₂) + ... + 4f(xₙ₋₁) + f(xₙ)]
Where f(x) = (x⁵ + 7)-1/3 for the example, n is even, and xi = a + i(b-a)/n
Enter function and limits, then click Calculate to see result.
Simpson's Rule is a numerical integration technique that approximates definite integrals by fitting parabolic curves through sets of three points. It's more accurate than the Trapezoidal Rule because it uses quadratic approximations instead of linear ones.
The method divides the integration interval into an even number of subintervals and applies the formula with alternating coefficients (1, 4, 2, 4, 2, ..., 4, 1) to achieve higher precision.
- Enter your function f(x) using standard mathematical notation
- Set the lower limit (a) and upper limit (b) of integration
- Choose an even number of subintervals (n) - more subintervals give higher accuracy
- Click "Calculate Integral" to get the approximate result
- Use "Show Steps" to see the detailed calculation process
∫ab f(x)dx ≈ (Δx/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 4f(xn-1) + f(xn)]
Where:
• Δx = (b - a) / n (subinterval width)
• n = even number of subintervals
• Coefficients: 1 for endpoints, 4 for odd indices, 2 for even indices
Problem: Approximate ∫₀¹ 1/(x⁵ + 7) dx using n = 4
Solution:
1. Δx = (1 - 0) / 4 = 0.25
2. Points: x₀ = 0, x₁ = 0.25, x₂ = 0.5, x₃ = 0.75, x₄ = 1
3. Function values: f(0) ≈ 0.142857, f(0.25) ≈ 0.142653, f(0.5) ≈ 0.141844, f(0.75) ≈ 0.139665, f(1) = 0.125
4. Apply coefficients [1, 4, 2, 4, 1]: Sum ≈ 1.133
5. Result: (0.25/3) × 1.133 ≈ 0.094417
What is Simpson's Rule?
Simpson's Rule is a numerical method for approximating definite integrals using parabolic approximations, providing higher accuracy than linear methods like the Trapezoidal Rule.
Why must n be even?
Simpson's 1/3 rule requires pairing adjacent intervals to form parabolic segments. Each parabola needs three points, so we need an even number of intervals to create complete pairs.
How accurate is Simpson's Rule?
Simpson's Rule has an error of O(h⁵), making it much more accurate than the Trapezoidal Rule's O(h³) error. It's exact for polynomials up to degree 3.
Difference between Trapezoidal Rule and Simpson's Rule?
The Trapezoidal Rule uses linear approximations (straight lines) while Simpson's Rule uses quadratic approximations (parabolas), resulting in significantly higher accuracy for smooth functions.