The area under sin x from 0 to π is exactly 2…

Check. The error of the trapezoid rule is about $1.66 / n^2$ (from the lesson: 0.104 for $n = 4$, so $0.104 \times 16 = 1.66$). The error is below 0.001 when $n^2 > 1660$, so $n > 40.7$, and the smallest whole number is 41.