Question

Let $f\left(x\right)$ be a continuous function such that the area bounded by the curve $y=f\left(x\right)$, x-axis and the lines $x=0$ and $x=a$ is $\frac{a^2}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a$, then $f\left(\frac{\pi }{2}\right)=$

• A. $1$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. none of these

Answer 1

The correct option is B $\frac{1}{2}$

We have, ${\int }_0^{a}f\left(x\right)dx=\frac{a^2}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a$

Differentiating w.r.t $a$, we get

$f\left(a\right)=a+\frac{1}{2}(\sin a+a\cos a)-\frac{\pi }{2}\sin a$

Substutute $a=\frac{\pi }{2};f\left(\frac{\pi }{2}\right)=\frac{\pi }{2}+\frac{1}{2}-\frac{\pi }{2}=\frac{1}{2}$