Question

I: The area bounded by the curve $y={\log }_{e}x,X$-axis and the line $x=e$ is 1 sq. units.
II: The area bounded by $X$-axis, the curve $y=f\left(x\right)$ and the lines $x=1,x=b$ is $\sqrt{b^2+1}-\sqrt{2}$ for all $b>1$ then $f\left(x\right)=\frac{x}{\sqrt{x^2+1}}$

Which of the above statement is correct?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II.

Answer 1

The correct option is C Both I and II

I) Area $={\int }_1^e\log xdx={\left[x\log x\right]}_1^e-{\int }_1^e\frac{x.1}{x}dx=e-{\left[x\right]}_1^e=1$
II) Area $={\int }_{x=1}^{b}f\left(x\right)dx=\sqrt{b^2+1}-\sqrt{2}$
$=\sqrt{b^2+1}-\sqrt{1+1}={\left[\sqrt{x^2+1}\right]}_1^b$
$\Rightarrow f\left(x\right)=\frac{d}{dx}\left(\sqrt{x^2+1}\right)=\frac{x}{\sqrt{x^2+1}}$