Question

Find the area of a curve $xy=4$ bounded by the lines $x=1$ and $x=3$ and $x-axis$

• A. $\cos \left(12\right)$
• B. $\log \left(64\right)$
• C. $\log \left(81\right)$
• D. $\log \left(27\right)$

Answer 1

The correct option is C

$\log \left(81\right)$

Explanation for correct option:

Step-1: Draw the graph with the help of given data.

Given, $x=1,x=3,x-axis$

and $xy=4$

$\Rightarrow$ $y=\frac{4}{x}.........\left(i\right)$

Step-2: Finding Required area.

Area bounded by the curve $x=1,x=3,y=\frac{4}{x}\&x-axis$ is ${\int }_1^3\frac{4}{x}dx$.

$=4{\int }_1^3\frac{1}{x}dx$

$=4{\left[\log \left(x\right)\right]}_1^3\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{(}\mathbf{\because }\mathbf{\int }\frac{\mathbf{1}}{\mathbf{x}}\mathbf{d}\mathbf{x}\mathbf{=}\mathbf{l}\mathbf{o}\mathbf{g}\left(x\right)\mathbf{)}$

$=4\left[\log \left(3\right)-\log \left(1\right)\right]$

$=4\log \left(3\right)\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{(}\mathbf{\because }\mathbf{l}\mathbf{o}\mathbf{g}\left(1\right)\mathbf{=}\mathbf{0}\mathbf{)}$

$=\log \left(3^4\right)\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{(}\mathbf{\because }\mathbf{l}\mathbf{o}\mathbf{g}\left(x^a\right)\mathbf{=}\mathbf{a}\mathbf{l}\mathbf{o}\mathbf{g}\left(x\right)\mathbf{)}$

$=\log \left(81\right)$

Hence, correct answer is option $\left(C\right)$