Subtopic: Thermodynamic principles of I.C. engines
Question Statement:
Values of the function y = 1/(1+x²) are y(0)=1; y(1)=0.5; y(2)=0.2; y(3)=0.1; y(4)=0.0588; y(5)=0.0385; y(6)=0.027. Using Simpson's one-third rule, the value of the integral ∫₀⁶ dx/(1+x²) is _____.
Official Answer Key:
1.35 to 1.40
Step-by-Step Solution & Explanation
Simpson's 1/3 rule with h = 1 and 6 equal intervals (7 ordinates y₀…y₆):
∫ ≈ (h/3)×[(y₀+y₆) + 4(y₁+y₃+y₅) + 2(y₂+y₄)]
= (1/3)×[(1+0.027) + 4(0.5+0.1+0.0385) + 2(0.2+0.0588)]
= (1/3)×[1.027 + 4(0.6385) + 2(0.2588)]
= (1/3)×[1.027 + 2.554 + 0.5176] = (1/3)× 4.0986 = 1.366. This lies within the accepted range 1.35–1.40. (The exact analytical value, tan⁻¹(6) = 1.4056 rad, is slightly higher — the Simpson approximation under-estimates it slightly because the tabulated y-values were themselves rounded.)
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