Question

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

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

Answer 1

The correct option is B

$\frac{1}{4}+{\log }_{\mathrm{e}}2$

Explanation for correct option

Calculating the area of the region bounded by the three curves

Given, the curves are $\mathrm{y}={\mathrm{x}}^3,\mathrm{y}=\frac{1}{\mathrm{x}},\mathrm{x}=2$

Visualizing the three curves on a graph paper:

Area of the required region is:
$$\begin{gathered} =\left(OMPNO\right) \\ ={\int }_0^1{x}^{3}dx+{\int }_1^2\frac{1}{x}dx \\ ={\left[\frac{x^4}{4}\right]}_0^1+[{\log }_{e}x{]}_1^2 \\ =\frac{1}{4}+{\log }_{e}2 \end{gathered}$$
Hence, the correct answer is Option (B)