I don't understand what my limits of integration will be after reversing the order of this double integral;
integral(from x=0 to x=3) integral(from y=sqrt(x/3) to y=1) of e^(y^3) dydx
$\displaystyle R=\left\{ (x,y)\in {{\mathbb{R}}^{2}}|0\le x\le 3,\,\sqrt{\frac{x}{3}}\le y\le 1 \right\}.$
You can describe the same region by just making a sketch. Or try to prove that $\displaystyle 0\le x\le 3{{y}^{2}}\le 3$ so $\displaystyle R=\left\{ (x,y)\in {{\mathbb{R}}^{2}}|0\le x\le 3{{y}^{2}},\,0\le y\le 1 \right\}.$