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Q.
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
KVPYKVPY 2019
Solution:
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
$A=\left|\int\limits_{1}^{2}\left\{\left(\frac{\sin\,x}{x}+e^{x}\right)-\left(\frac{\sin\,x}{x}+\frac{x^{2}}{2}\right)\right\}dx\right|$
$=\left|\int\limits_{1}^{2}\left(e^{x}-\frac{x^{2}}{2}\right)dx\right|=\left|\left[e^{x}-\frac{x^{3}}{6}\right]_{1}^{2}\right|$
$=e^{2}-e-\frac{7}{6}$