Evaluate a double integral over a domain

Question

I'm asked to evaluate $$\iint(x^3+y)$$ over the ellipse on the xy plane such that $2x^2+y^2<2y$

I figured that the ellipse can be parametrized by $$\vec r(t)=\left(\frac{\cos t}{\sqrt2};1-\sin t\right)$$ but I dont know what to do after.

Thanks in advance

Answer 1

$2x^2+y^2-2y < 0 \to 2x^2+(y-1)^2 < 1 \to \dfrac{x^2}{\left(\frac{1}{\sqrt{2}}\right)^2}+\dfrac{(y-1)^2}{1^2} <1 \to A = \displaystyle \int_{-\frac{1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}\int_{1-\sqrt{1-2x^2}}^{1+\sqrt{1-2x^2}}(x^3+y)dydx$