Question

Find the area bounded by the y-axis and the curve $x=e^y\sin \pi y$, $y=0$, $y=1$.

• A. $\frac{\pi (e-1)}{1-{\pi }^2}$
• B. $\frac{\pi (e-1)}{1+{\pi }^2}$
• C. $\frac{\pi (e+1)}{1-{\pi }^2}$
• D. $\frac{\pi (e+1)}{1+{\pi }^2}$

Answer 1

The correct option is D $\frac{\pi (e+1)}{1+{\pi }^2}$

$A={\int }_0^1{e}^y\sin \pi ydy$

Integration by parts

$U=\sin \pi y$ , $V=R^y$

$du=\pi \cos \pi y$ ,

$e^y\sin \pi y-\pi {\int }_0^1\left(\cos \pi y\right)e^{y}dy$

$du=\pi \sin \pi y$ $\int v=e^y$

$A=e^y\sin \pi y-\pi \cos \pi y{e}^y+{\pi }^2{\int }_0^1{e}^y\sin \pi ydy$

$A(1-{\pi }^2)=[e^y\sin \pi y-\pi \cos \pi y{e}^y{]}_0^1$

$A=\frac{1}{1-{\pi }^2}(\pi e^1+\pi )=\frac{\pi (e+1)}{1-{\pi }^2}$