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Area between Two Curves

Consider two ...

Question

Consider two curves $C_{1} : y = \frac{1}{x}$ and $C_{2} : y = l n x$ on the $x y$ 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 sum of logarithm of possible values of $a$ is:

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Solution

$D_{1} = D_{2} \Rightarrow \int_{1}^{c} (\frac{1}{x} - ln x) d x = \int_{c}^{a} (ln x - \frac{1}{x}) d x \Rightarrow \int_{1}^{a} (\frac{1}{x} - ln x) d x = 0 \Rightarrow {[x - (x - 1) ln x]}_{1}^{a} = 0 \Rightarrow a - (a - 1) ln a - 1 - 0 = 0 \Rightarrow (a - 1) ln a = a - 1 \Rightarrow a = 1$

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