I am practising setting up limits of integration for double integrals. I'm struggling with the question below, so I was hoping to get some feedback on my current solution (for the limit portion anyway). I have been tasked with integrating the following:
$\iint_{R}^{}x\cos{y} dA$
Where $R$ is the region bounded by the curves $y=3x^2-1, y=x^2,y=0$. I must integrate first with respect to $x$, then $y$, and then with respect to $y$, then $x$.
So far, I have the following:
$\int_{0}^{\frac{1}{\sqrt{2}}}\int_{\sqrt{\frac{y+1}{3}}}^{\sqrt{y}}x\sin{y}dxdy$
And
$\int_{-\frac{1}{\sqrt{3}}}^{0}\int_{0}^{x^2} x\sin{y}dydx+ \int_{0}^{\frac{1}{\sqrt{3}}}\int_{0}^{3x^2-1} x\sin{y}dydx$
If anyone could provide feedback on this, it would be greatly appreciated!
