The area of the region enclosed by the curve $f(x)=\max\{\sin x,\cos x\},\ -\pi\leq x\leq\pi$ and the $x$-axis is
Step-by-Step Solution
Key Concept: $\max\{\sin x,\cos x\}$ alternates between $\sin x$ and $\cos x$ at $x=-\frac{3\pi}{4},\ -\frac{\pi}{4},\ \frac{\pi}{4}$. Integrate the absolute value over each sub-interval.
Total area $=\left|\int_{-\pi}^{-3\pi/4}\sin x\,dx\right|+\left|\int_{-3\pi/4}^{\pi/4}\cos x\,dx\right|+\left|\int_{\pi/4}^{\pi}\sin x\,dx\right|=4$.
Correct Answer: 2