What is the total area between $y = \sin x$ and the $x$-axis from $x = 0$ to $x = 2\pi$?

Explanation

1. 01

   The curve crosses the axis at $x = \pi$: above it on $(0, \pi)$, below it on $(\pi, 2\pi)$. Take the two parts separately.

2. 02

   $\int_0^{\pi} \sin x\,dx = 2$ and $\left|\int_{\pi}^{2\pi} \sin x\,dx\right| = 2$, so the total area is 4 square units.