Project ForgeFunction f is known at the following points: x=0,0.3,0.6,0.9,1.2,1.5,1.8,2.1,2.4,2.7,3.0 with f(x)=0,0.09,0.36,0.81,1.44,2.25,3.24,4.41,5.76,7.29,9.00 respectively. What is the value of ∫[0 to 3] f(x) dx computed using the trapezoidal rule?
Explanation
These points are exactly f(x)=x² sampled at h=0.3 intervals (0.3²=0.09, 0.6²=0.36, etc.). The trapezoidal rule gives ∫≈(h/2)[f₀+2(f₁+...+f₉)+f₁₀] = (0.3/2)[0 + 2×(0.09+0.36+0.81+1.44+2.25+3.24+4.41+5.76+7.29) + 9.00] = 0.15×[2×25.65+9.00] = 0.15×60.30 = 9.045. (The true integral of x² from 0 to 3 is exactly 9 — the trapezoidal rule's slight overestimate, 9.045, is expected since x² is convex.)
Source: GATE CS 2013 (2013)
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