The area bounded by y = ln x, the x-axis and the ordinates x = 0 and x = 1, is

1

The required area is shown shaded in the figure, and area =$-\int\limits_0^1y\,dx$.

$=-\int\limits_0^1ln\,x\,dx=(x-x\,ln\,x)_0^1=1$

Hence (A) is the correct answer.

Alternative:

The equation of the given curve may be written as x = e^y so that

area = $\int\limits_0^{-∞}x\,dy=\int\limits_0^{-∞}e^y\,dy=e^y|_0^{-∞}=1$.