examday

/TMUA

Let $k$ be an integer. Approximating

using the trapezium rule with $n$ strips results in an underestimation.

Which of the following must be true?

• I. Approximating $\displaystyle\int_{(2k-1/4)\pi}^{(2k+1/4)\pi}\dfrac1{\cos x}\dd x$ using the trapezium rule with $n$ strips results in an overestimation.
• II. $\displaystyle\int_{(k-1/4)\pi}^{(k+1/4)\pi}\dfrac1{\cos^3x}\dd x>\int_{(k-1/4)\pi}^{(k+1/4)\pi}\dfrac1{\cos x}\dd x$.
• III. $-\dfrac\pi{\sqrt2}<\displaystyle\int_{(k-1/4)\pi}^{(k+1/4)\pi}\dfrac1{\cos x}\dd x<0$.