Question

Consider two curves $C_1:y=\frac{1}{x}$ and $C_2$ : $y=lnx$ on the xy plane Let $D_1$ denotes the region surrounded by $C_1$, $C_2$ and the line $x=1$ and $D_2$ denotes the region surrounded by $C_1$, $C_2$ and the line $x=a$ If $D_1$=$D_2$ then the value of 'a':

• A. $\frac{e}{2}$
• B. $e$
• C. $e-1$
• D. $2(e-1)$

Answer 1

The correct option is B $e$

Let the x point of intersection of the two curves be (k)
$D1={\int }_1^k(\frac{1}{x}-\ln x)dx$
$D2={\int }_k^a(-\frac{1}{x}\ln x)dx$
$D1+D2=0$
${\int }_1^a(\frac{1}{x}dx={\int }_1^a(\ln x)dx$
$\ln a=a\ln a-a+1$
$(\ln a-1)(1-a)=0$
$a=e$