While solving a simple problem for finding are of the region bounded by $x=y^2$ and $x=y$. Are the following correct limits?

When $x$ is the outer integration variable $$\int^1_0\int^\sqrt{x}_xdydx$$

When $y$ is the outer integration variable?


The reason for asking the question is, I am getting the different answer for both the cases which should not be the case.


\begin{align} \int_0^1 \int_x^{\sqrt{x}} \, \, dydx &= \int_0^1 x^\frac12-x \, dx \\ &= \frac23 -\frac12 \\ &= \frac16 \end{align}

\begin{align} \int_0^1 \int_{y^2}^{y} \, \, dxdy &= \int_0^1 y-y^2 \, dx \\ &= \frac12 -\frac13 \\ &= \frac16 \end{align}

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