\(\int_{0}^{100\pi}\displaystyle\left|\sin\frac{x}{2}\right|dx =\)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 200
\(\sin\frac{x}{2}\) has period \(4\pi\), and taking the modulus makes the negative half equal to the positive half, so \(\left|\sin\frac{x}{2}\right|\) has period \(2\pi\). The interval \([0, 100\pi]\) is 50 such periods:
\( \displaystyle \int_{0}^{100\pi}\left|\sin\frac{x}{2}\right|dx = 50\int_{0}^{2\pi}\left|\sin\frac{x}{2}\right|dx \)
For \(0 \le x \le 2\pi\), \(\frac{x}{2}\) lies in \([0, \pi]\), where the sine is not negative:
\( \displaystyle \int_{0}^{2\pi}\sin\frac{x}{2}\,dx = \left[-2\cos\frac{x}{2}\right]_{0}^{2\pi} \)
\(= -2\cos\pi+2\cos 0\)
\(= 2+2 = 4\)
Hence the integral is \(50 \times 4 = 200\).