Question

Area bounded by the curve $y=\ln x,y=0$ and $x=3$ is

• A. $(\ln 9-2)$.sq.unit
• B. $(\ln 27-2)$.sq.unit
• C. $\ln \left(\frac{27}{e^2}\right)$.sq.unit
• D. $>3$.sq.unit

Answer 1

Answer: (B), (C)

The correct options are
B $(\ln 27-2)$.sq.unit
C $\ln \left(\frac{27}{e^2}\right)$.sq.unit

Required Area$={\int }_1^3\ln xdx$
$={[x\ln x-x]}_1^3$
$=(3\ln 3-2)$
$=\ln 27-2$.sq.unit(option $\left(b\right)$ is correct)
$=\ln 27-\ln e^2$
$=\ln \left(\frac{27}{e^2}\right)$.sq.unit(option $\left(c\right)$ is correct)
Also, $\ln \left(\frac{27}{e^2}\right)>3$
$\Rightarrow \ln \left(\frac{27}{e^2}\right)>3\ln e$
$\Rightarrow \ln 27-\ln e^2>3\ln e$
$\Rightarrow \ln 27-2\ln e>3\ln e$
$\Rightarrow \ln 27>3\ln e+2\ln e$
$\Rightarrow \ln 27>5\ln e$
$\Rightarrow \ln 27>\ln e^5$ which is false.