What is \varphi(a) equal to ?
Consider the following for the next two (02) items that follow : Let \varphi(a)=\int_{a}^{a+100\pi}|\sin x|\,dx
- A. 0
- B. a
- C. 100a
- D. 200 ✓
Correct Answer: D. 200
Explanation
The function |\sin x| is periodic with period \pi. The integral of a periodic function with period T over any interval of length nT is n \times \int_0^T f(x)dx. Since the integration length is 100\pi, which is exactly 100 periods, we compute 100 \int_0^\pi |\sin x|dx = 100 \int_0^\pi \sin x\,dx. Since \int_0^\pi \sin x\,dx = 2, the value is 100 \times 2 = 200.
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