I am trying to Integrate the area mentioned below using ImplicitRegion function, following [this][1] resource. This method worked in many other scenario.
The plot was accurate, but the integration returned 1/10 which is not true as is verified by the traditional integration.
I have tried integration over $-1\leq y\leq 0$ and $0\leq y\leq 1$ separately and also using Surd[x,3]^2 but things got much weirder. Can any one give a hint how I can improve or correct this?

$\pmb{\text{reg}=\text{ImplicitRegion}\left[y\leq x\leq y^{\frac{2}{3}},\{x,\{y,-1,1\}\}\right]}$



$\text{ImplicitRegion}\left[y\leq x\leq y^{2/3}\&\&-1\leq y\leq 1,\{x,y\}\right]$



$\pmb{\text{RegionPlot}[\text{reg}]}$
[![enter image description here][2]][2]


$\pmb{\text{Integrate}[1,y\in \text{reg}]}$



$\frac{1}{10}$



$\pmb{\int_{-1}^1 \left(y^{\frac{2}{3}}-y\right) \, dy}$



$\frac{3}{5} \left(1+(-1)^{2/3}\right)$


  [1]: https://www.wolfram.com/mathematica/new-in-10/basic-and-formula-regions/integrate-over-regions.html
  [2]: https://i.sstatic.net/cbyuf.jpg