Question

Consider the region formed by the lines $x=0,y=$ $0,x=2,y=2$. Area enclosed by the curves $y=e^x$ and $y=\ln x$, within this region is removed, then the area of the remaining region is:

• A. $2(1+2{\ln }^{2}2)$
• B. $2(2\ln 2-1)$
• C. $(2\ln 2-1)$
• D. $1+2{\ln }^{2}2$

Answer 1

The correct option is B

$2(2\ln 2-1)$

Area $={\int }_1^2\ln xdx+{\int }_1^2\ln ydy=2{\int }_1^2\ln x=2{[x\ln x-x]}_1^2=2[2\ln 2-2+1]=4\ln 2-2=2(2\ln 2-1)$