Question

The area bounded by the x-axis, the curve \(y=f(x)\) and the lines \(x=1,\;x=b\) is equal to \(\sqrt{b^2+1}-\sqrt{2}\) for all \(b > 1\). Then \(f(x)\) is

• A. \(\sqrt{x-1}\)
• B. \(\sqrt{x+1}\)
• C. \(\sqrt{x^2+1}\)
• D. \(\dfrac{x}{\sqrt{1+x^2}}\)

Hint:

Differentiate area function to recover the integrand.

Answer 1

The Correct Option is D

Step 1: \[ \int_1^b f(x)\,dx=\sqrt{b^2+1}-\sqrt{2} \]
Step 2: Differentiate w.r.t. \(b\): \[ f(b)=\frac{b}{\sqrt{1+b^2}} \]