Question

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

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

Answer 1

The correct option is A

$1$ sq units

Explanation for the correct option:

Step 1. Draw the graphs of $y={\log }_{e}x,x=0,y\le 0$:

Given that $y={\log }_{e}x,x=0,y\le 0$ and $x$-axis

Now, we can see intersecting point of $y={\log }_{e}x$ and $x$-axis is $\left(1,0\right)$

Step 2. Area bounded by the curves $=\left|{\int }_0^1\left(0-{\log }_{e}x\right)dx\right|$

$$\begin{gathered} =\left|{\int }_0^1{\log }_{e}xdx\right| \\ =\left|\left[x{\log }_{e}x\right]-{\int }_0^1\frac{1}{x}xdx\right| \\ ={\left|x{\log }_{e}x-x\right|}_0^1 \\ =1 \end{gathered}$$

$\therefore$Area bounded by the curves is $\mathbf{1}$ sq unit

Hence, Option ‘A’ is Correct.