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

Answer: (B)

The area of region bounded by the curves
$x\left(y-e^x\right)=\sin x$
or $y=\frac{\sin x}{x}+e^x$ and
$2xy=2\sin x+x^3$
or $y=\frac{\sin x}{x}+\frac{x^2}{2}$
between lines $x=1$ and $x=2$ is
$A=\left|\underset{1}{\overset{2}{\int }}\left\{\left(\frac{\sin x}{x}+e^x\right)-\left(\frac{\sin x}{x}+\frac{x^2}{2}\right)\right\}dx\right|$
$=\left|\underset{1}{\overset{2}{\int }}\left(e^x-\frac{x^2}{2}\right)dx\right|=\left|{\left[e^x-\frac{x^3}{6}\right]}_1^2\right|$
$=e^2-e-\frac{7}{6}$