GATE 2020 AG – Question 46
Use Simpson’s one-third rule with six equal intervals of width π/12 to approximate $\int_0^{\pi/2}\sqrt{\cos\theta}\,d\theta$ (three decimals).
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Correct answer: 1.186 to 1.189
Explanation
Put $h=\pi/12$ and $f_j=\sqrt{\cos(jh)}$. Set $A=f_1+f_3+f_5$ and $B=f_2+f_4$. Simpson’s six-interval formula is $$I\approx\frac h3[f_0+f_6+4A+2B].$$ The ordinates are approximately 1, 0.982815, 0.930605, 0.840896, 0.707107, 0.508743 and 0. Substitution gives 1.1873, or 1.187 to three decimals. This is the requested numerical approximation, rather than the exact integral.