Question

Let $f\left(x\right)$ be a continuous and positive function and the area bounded by $y=f\left(x\right),$ the $x$ -axis and the lines $x=0$ and $x=a$ is $4a^{3}+sin a$ sq. units $\left(a > 0\right)$ . If $f\left(\frac{\pi }{2}\right)=k\left(\pi \right)^{2},$ then the value of $k$ is

Answer 1

Given area is $\displaystyle \int _{0}^{a} f\left(x\right)dx$
$\therefore \displaystyle \int _{0}^{a} f\left(x\right)dx=4a^{3}+sin a$
Differentiating w.r.t. $a,$ we get
$f\left(a\right)=12a^{2}+cos a$
$\Rightarrow f\left(\frac{\pi }{2}\right)=3\left(\pi \right)^{2}$
$\Rightarrow k=3$