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Q.
Area bounded by the curve $y = log_ex,\, x = 0,\, y \le 0$, and x-axis is :
Application of Integrals
Solution:
Given, curves are $y = log_ex,\, x = 0,\, y \le 0$ and x-axis.
$\therefore $ The given shaded region shows the required area
Now, Intersecting point of $y = log_ex$ & x-axis is (1, 0)
$\therefore $ Required area
$= \left|\int\limits^{1}_{0}\left(0-log_{e}\,x\right)dx\right| = \left|\int\limits^{1}_{0}log_{e}\,x\,dx\right|$
$= \left|x \,log_{e}\,x]^{1}_{0}-\int\limits^{1}_{0} \frac{1}{x}\,x\,dx\right|$
$= \left|x \,log_{e}\,x]^{1}_{0}-x]^{1}_{0}\right|$
$= \left[\left|x \,log_{e}\,x - x\right|\right]^{1}_{0} = 1$ sq. unit
