Hi all,
I'm wondering if someone can point me in the right direction for integrating this integral...
$\displaystyle \int \sin{(\frac{x}{y})} dy$
I'm not sure where to start,
thanks all.
If anyone could check through my working it would be great.
Q. Evaluate the following double integral
$\displaystyle \int\int_R \sin{\frac{x}{y}} dA $
Where R is the region bounded by the y-axis, $\displaystyle y = \pi$, $\displaystyle x = y^2$
Drawing a horizontal line throughout the region, we end up with...
$\displaystyle \int^{\pi}_0 \int^{y^2}_0 \sin{\frac{x}{y}} dxdy $
This turns through integration into
$\displaystyle \int^{\pi}_0 [ -y\cos{\frac{x}{y}}]^{y^2}_0 dy $
Subbing in values turns it into
$\displaystyle \int^{\pi}_0 [-y\cos{y} + y] dy $
Integration by parts turning into
$\displaystyle [\frac{y^2}{2} - \cos{y} - y\sin{u}]^{\pi}_0 $
Giving
$\displaystyle \frac{\pi^2}{2} - \cos{\pi} - \pi\sin{\pi} + \cos{0} $
Am I right so far?
Thanks.