- 2069 Bhadra · 6 marks
Romberg: e^(-x²) from 1 to 2
Short answer
Build trapezoidal values for h = 1, 0.5, 0.25, 0.125 and apply Richardson extrapolation. The successive results agree to three decimals, giving the integral approximately 0.135.
Evaluate ∫ from 1 to 2 of e^(-x²) dx using Romberg method correct up to 3 decimal places.
Romberg integration: build trapezoidal values T(h) for h = 1, 0.5, 0.25, 0.125, then extrapolate with R(i,j) = [4ʲ·R(i,j−1) − R(i−1,j−1)] / (4ʲ − 1), until two successive values agree to 3 decimals.
f(x) = e^(−x²), a = 1, b = 2.
Romberg table (6 decimals kept for accuracy)
| h | T (order h²) | R1 | R2 | R3 |
|---|---|---|---|---|
| 1 | 0.193098 | |||
| 0.5 | 0.149248 | 0.134632 | ||
| 0.25 | 0.138720 | 0.135210 | 0.135249 | |
| 0.125 | 0.136121 | 0.135254 | 0.135257 | 0.135257 |
Check: R2 at h = 0.25 is 0.135249 and R3 at h = 0.125 is 0.135257; both round to 0.135, so the required 3-decimal accuracy is reached.
Answer: ∫₁² e^(−x²) dx ≈ 0.135 (exact value 0.135257).
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Romberg: (e^x + sin x)/(1+x²) · Numerical Method, chapter 5 · asked 2×
- Romberg: 1/(1+x²) from -1 to 1 · Numerical Method, chapter 5
- Romberg: 1/x² from 1 to 2 · Numerical Method, chapter 5
- Evaluate ∫∫ e^(y/x) dy dx · Mathematics, chapter 2
- Evaluate ∫∫ e^(x²) dx dy · Mathematics, chapter 2
- Evaluate ∫ dx/x³ from ∞ to α · Mathematics I, chapter 2
- Choose correct words (grammar) 2073 Shrawan · Communication English, chapter 3