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Gerard Sayson TMUA Mock Paper 1 · Q5

Paper 1CalculusTrigonometry

Use the trapezium rule involving four points equally spaced apart to approximate the integral

∫02π1+sin⁡2x dx\int_0^{2\pi}\sqrt{1+\sin^2x}\,\mathrm{d}x

and determine whether there is an overestimate or underestimate.

A2π3(1+1+32+1−32)\dfrac{2\pi}{3}\left(1+\sqrt{1+\dfrac{\sqrt3}{2}}+\sqrt{1-\dfrac{\sqrt3}{2}}\right), and there is an underestimate
B2π3(1+1+32+1−32)\dfrac{2\pi}{3}\left(1+\sqrt{1+\dfrac{\sqrt3}{2}}+\sqrt{1-\dfrac{\sqrt3}{2}}\right), and there is an overestimate
C2π(1+7)3\dfrac{2\pi(1+\sqrt7)}{3}, and there is an underestimate
D2π(1+7)3\dfrac{2\pi(1+\sqrt7)}{3}, and there is an overestimate
Eπ(1+2)\pi(1+\sqrt2), and there is an underestimate
Fπ(1+2)\pi(1+\sqrt2), and there is an overestimate

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