Question

Find the area enclosed between the curves $y=\ell n(x+e),$ $x=\ell n\left(\frac{1}{y}\right)$ .

Answer 1

R.E.F image

Given curves $y=ln(x+e)$

$x=ln\left(\frac{1}{y}\right)$

$\to e^x=1/y$

$\to y=e^{-x}$

Area $={\int }_0^1(x_1-x_2)dy={\int }_0^1[-1ny-(e^y-e\left)\right]dy$

$-{\int }_0^1\ln ydy-{\int }_0^1{e}^{y}dy+{\int }_0^{1}edy$

$=-[y\ln y-y{]}_0^1-[e^y{]}_0^1+e[y{]}_0^1$

$=1-e+1+e$

$=2$ sq.units