Question

What is the area bounded by $y=\log x,x-axis$ and ordinates $x=1,x=2$?

• A. $\frac{1}{2}{\left(\log x\right)}^2$
• B. $\log \left(\frac{2}{e}\right)$
• C. $\log \left(\frac{4}{e}\right)$
• D. $\log 4$

Answer 1

The correct option is C

$\log \left(\frac{4}{e}\right)$

Explanation of correct option:

Finding area within the curves:

Given, $y=\log x$

Plotting the curves,

Here, blue line is $y=\log x,$ green line is $x=1$ and orange line is $x=2$.

And, the area bounded by these three curves and X-axis is the area within them

Therefore, the required area can be expressed as,

$$\begin{gathered} A={\int }_1^2\log xdx \\ ={\left[x\log x\right]}_1^2-{\int }_1^2\frac{x}{x}dx\mathbf{}\mathbf{\left(}\mathbf{\text{using integration by parts}}\mathbf{\right)} \\ ={\left[x\log x-x\right]}_1^2 \\ =2\log 2-2+1 \\ =2\log 2-1 \\ =\log 4-1\mathbf{}\left(\because alogb=log{b}^a\right) \\ =\log 4-\log e\mathbf{(}\mathbf{\because }\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{e}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{)} \\ =\log \left(\frac{4}{e}\right)\left(\because loga-logb=log\left(\frac{a}{b}\right)\right) \end{gathered}$$

Hence, the area within curve is $\log \left(\frac{4}{e}\right)$.

Hence, option (C) is the correct answer.