Question

The area of the region bounded by the curves $y=x^3,y=\frac{1}{x},x=2$ is

• A. $4-{\log }_{e}2$
• B. $\frac{1}{4}+{\log }_{e}2$
• C. $3-{\log }_{e}2$
• D. $\frac{15}{4}-{\log }_{e}2$

Answer 1

The correct option is B $\frac{1}{4}+{\log }_{e}2$

First of all we draw the graph,
$y=x^3,y=\frac{1}{x},x=2$
$\therefore$ Required area i.e., $\left(OMPNO\right)$
$={\int }_0^1{x}^{3}dx+{\int }_1^2\frac{1}{x}dx$

$={\left[\frac{x^4}{4}\right]}_0^1+[{\log }_{e}x{]}_1^2=\frac{1}{4}+{\log }_{e}2$