Let φ(a) = ∫_a^(a+100π) |sin x| dx. What is φ(a) equal to?

|sin x| has period π, and ∫_0^π |sin x| dx = ∫_0^π sin x dx = [-cos x]_0^π = 1 + 1 = 2. Over an interval of length 100π, which contains exactly 100 full periods, the integral equals 100 × 2 = 200, regardless of the starting point a. Hence φ(a) = 200.