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Identify and explain why a statement is false

by RoRi

Given two statements:

  1. The integral

        \[ \int_{2 \pi}^{4 \pi} \frac{\sin t}{t} \, dt > 0 \]

    because

        \[ \int_{2 \pi}^{3 \pi} \frac{\sin t}{t} \, dt > \int_{3 \pi}^{4 \pi} \frac{|\sin t|}{t} \, dt. \]

  2. The integral

        \[ \int_{2 \pi}^{4 \pi} \frac{\sin t}{t} \, dt = 0 \]

    because, by the weighted mean value theorem (Theorem 3.16 in Apostol), there exists a c in [2 \pi, 4 \pi] such that

        \[ \int_{2 \pi}^{4 \pi} \frac{ \sin t}{t} \, dt = \frac{1}{c} \int_{2 \pi}^{4 \pi} \sin t \, dt = \frac{ \cos (2\pi) - \cos (4 \pi)}{c} = 0. \]

    3. Identify which of the two statements is false, and explain why.

Statement (b) is false since we may not apply the weighted mean value theorem in this case. The weighted mean value theorem requires the function g(t) to not change sign on the interval [2 \pi, 4 \pi]. Since g(t) = \sin t does change sign on the interval, we cannot apply the theorem.

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