Question

If the area enclosed by the curves $f(x)={\cos }^{-1}(\cos x)$ and $g(x)={\sin }^{-1}(\cos x)$ in $x\in \left[\frac{9\pi }{4},\frac{15\pi }{4}\right]$ is $\frac{a{\pi }^2}{b}$ (where $a$ and $b$ are coprime), then find $(a+b)$.

Answer 1

Answer: 17

We have $g(x)={\sin }^{-1}(\cos x)=\frac{\pi }{2}-{\cos }^{-1}(\cos x)$
Both the curves bound the regions of same area
in $\left[\frac{\pi }{4},\frac{7\pi }{4}\right],\left[\frac{9\pi }{4},\frac{15\pi }{4}\right]$ and so on
$\therefore$ Required area $=$ area of shaded square $=\frac{9{\pi }^2}{8}=\frac{a{\pi }^2}{b}$
$\therefore a=9$ and $b=8$
Hence $a+b=17$