I have these particular exercise that i cannot solve. I know i have to change the variables, but i cannot figure out if i should use polar coords or any other change.
Let D be the region delimited by: $ D = \{(x,y) \in \mathbb{R} ^{2} : (x-1)y \geq 0, \frac{(x-1)^2}{9} + \frac{y^2}{25} \leq 1 \} $ Calculate:
$ \iint\limits_D \sin((x-1)^2 + \frac{9y^2}{25}) \,dxdy $
I've tried using $u = \frac{(x-1)}{3}$ and $v = \frac{y}{5}$ so that i can replace in the integral the following:
$ \iint\limits_D \sin(9(u^2 + v^2)) \frac{1}{15} \,dudv $
knowing the Jacobian is $J(x,y)=\frac{\partial (u,v)}{\partial (x,y)}=\frac{1}{15}$.
But i don't know where to follow, or if the variable changes i've made are correct. Can I use that $u^2 + v^2 = 1$, or that's just for polar coords?
Thanks a lot for your help!