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

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 $a + b =17$