Question

Area bounded by the curve $y=lo{g}_{e}x,x=0,y\le 0$, and x-axis is :

• A. 1 sq. unit
• B. $\frac{1}{2}$ sq. unit
• C. 2 sq. unit
• D. None of these

Answer 1

Answer: (A)

Given, curves are $y=lo{g}_{e}x,x=0,y\le 0$ and x-axis.
$\therefore$ The given shaded region shows the required area

Now, Intersecting point of $y=lo{g}_{e}x$ & x-axis is (1, 0)
$\therefore$ Required area
$=\left|\underset{0}{\overset{1}{\int }}\left(0-lo{g}_{e}x\right)dx\right|=\left|\underset{0}{\overset{1}{\int }}lo{g}_{e}xdx\right|$
$=\left|xlo{g}_{e}x{]}_0^1-\underset{0}{\overset{1}{\int }}\frac{1}{x}xdx\right|$
$=\left|xlo{g}_{e}x{]}_0^1-x{]}_0^1\right|$
$={\left[\left|xlo{g}_{e}x-x\right|\right]}_0^1=1$ sq. unit