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

1. Over $[0, 2\pi]$, $\cos x$ is positive on $\left[0, \frac{\pi}{2}\right]$ and $\left[\frac{3\pi}{2}, 2\pi\right]$, and negative on $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$. 2. Area $= \int_0^{\pi/2}\cos x\,dx + \left|\int_{\pi/2}^{3\pi/2}\cos x\,dx\right| + \int_{3\pi/2}^{2\pi}\cos x\,dx$. 3. Using $[\sin x]$, the three magnitudes are $1$, $2$ and $1$. 4. Adding, the area $= 1 + 2 + 1 = 4$. 5. The plain integral $\int_0^{2\pi}\cos x\,dx = 0$, so $0$ is the trap from skipping absolute values. _Source: NCERT Class 12 Mathematics Ch 8 "Application of Integrals", p.5_