Question

Let $f(x)$ be a continuous function such that the area bounded by the curve $y=f(x),x$ -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

Answer: (B)

We have, $\underset{0}{\overset{a}{\int }}f(x)dx=\frac{a^2}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a$
Differentiating w.r.t. a, we get
$f(a)=a+\frac{1}{2}(\sin a+a\cos a)-\frac{\pi }{2}\sin a$
Put $a=\frac{\pi }{2};f\left(\frac{\pi }{2}\right)=\frac{\pi }{2}+\frac{1}{2}-\frac{\pi }{2}=\frac{1}{2}$