The area bounded by the curve $y = \sin x$, the $x$-axis, between $x = 0$ and $x = 2\pi$ is

1. On $[0, 2\pi]$, $\sin x \geq 0$ for $[0, \pi]$ and $\sin x \leq 0$ for $[\pi, 2\pi]$. 2. Area $= \int_0^{\pi}\sin x\,dx + \left|\int_{\pi}^{2\pi}\sin x\,dx\right|$. 3. $\int_0^{\pi}\sin x\,dx = [-\cos x]_0^{\pi} = 2$; the second piece also has magnitude $2$. 4. Total area $= 2 + 2 = 4$. 5. Without absolute value the two halves cancel to $0$, the common trap. _Source: NCERT Class 12 Mathematics Ch 8 "Application of Integrals", p.6_