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Y and A Tutoring Paper 1 with Answers and Hints · Q4

Paper 1Logic & proofCalculus

A student uses the trapezium rule with 22 strips of equal width to estimate the value of the definite integral:

∫024−x2 dx.\int_0^2\sqrt{4-x^2}\,\dd x.

By comparing this trapezium rule estimate to the exact value of the integral, and considering the shape of the curve, which of the following inequalities is rigorously proven to be true?

Aπ<1+3\pi<1+\sqrt3
Bπ>1+3\pi>1+\sqrt3
Cπ<2+23\pi<2+2\sqrt3
Dπ>2+23\pi>2+2\sqrt3
Eπ=1+3\pi=1+\sqrt3

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