Question

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

Answer 1

Side $AB=\sqrt{{(\frac{9\pi }{4}-3\pi )}^2+{(\frac{\pi }{4}-\pi )}^2}$

$=\sqrt{{\left(\frac{3\pi }{4}\right)}^2+{\left(\frac{3\pi }{4}\right)}^2}$

$=\sqrt{2}\frac{3\pi }{4}$

Side $BC=\sqrt{{(\frac{15\pi }{4}-3\pi )}^2+{(\frac{\pi }{4}-\pi )}^2}$

$=\sqrt{2}\frac{3\pi }{4}$

Semi-Area $=\frac{1}{2}\sqrt{2}.\frac{3\pi }{4}.\sqrt{2}.\frac{3\pi }{4}$

$=\frac{9{\pi }^2}{16}$

Area $=\frac{9{\pi }^2}{8}$

$a=9,b=8$

$a-b=1$

Hence, answer is $1$.