Simpson's Rule Calculator
Calculate definite integrals using Simpson's Rule numerical approximation
Results
∫ = 2.033333
Calculation Steps
Initial Values
f(a) + f(b)5.000000
Odd Terms
4 × Σf(x₂ᵢ₋₁)40.000000
Even Terms
2 × Σf(x₂ᵢ)16.000000
Final Result
(h/3) × (sum + oddSum + evenSum)2.033333
Error Analysis
Absolute Error
1.667e+10
Relative Error
8.197e+9
Error Bound
1.667e+10
Recommendations
Accuracy
Current interval count provides good accuracy
Error Estimation
Consider using more intervals to reduce error
Function Complexity
Verify function is continuous on the interval
Interval Selection
Use even number of intervals for Simpson's rule
Understanding Simpson's Rule
Formula
∫[a→b] f(x)dx ≈ (h/3)[f(a) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + ... + f(b)]where h = (b-a)/n and n is the number of intervals (must be even)
Key Points
- Simpson's Rule provides better accuracy than the Trapezoidal Rule
- The number of intervals must be even
- Accuracy improves with more intervals
- Works best with smooth, continuous functions
Applications
- Engineering calculations
- Physics simulations
- Statistical analysis
- Numerical approximations