Question

The area of the region bounded by the lines $x=1,x=2,$ and the curves $x(y-e^x)=\sin x$ and $2xy=2\sin x+x^3$ is

• A. $e^2-e-\frac{1}{6}$
• B. $e^2-e-\frac{7}{6}$
• C. $e^2-e+\frac{1}{6}$
• D. $e^2-e+\frac{7}{6}$

Answer 1

The correct option is B $e^2-e-\frac{7}{6}$

$x(y-e^x)=\sin x\Rightarrow y=e^x+\frac{\sin x}{x}$

$2xy=2\sin x+x^3\Rightarrow y=\frac{x^2}{2}+\frac{\sin x}{x}$
Required area $=|\underset{1}{\overset{2}{\int }}(e^x+\frac{\sin x}{x})-(\frac{x^2}{2}+\frac{\sin x}{x})dx|=|\underset{1}{\overset{2}{\int }}{e}^x-\frac{x^2}{2}dx|=|{[e^x-\frac{x^3}{6}]}_1^2|=(e^2-e)-\left(\frac{8-1}{6}\right)=e^2-e-\frac{7}{6}$