What did I do wrong in this double integral calculation?

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I tried solving the double integral below, but I get the wrong answer. According to Symbolab, it should be $\frac{\sqrt{\pi}}{8}$. Can someone take a look at my calculation and tell me what I did wrong? The exercise tells to calculate the integral, giving a hint that it is way easier if we change the order of integration.

I change the range of integration like below because $0<y<1$ and $y^2<x<1$, so $0<y<\sqrt{x}<1$ and $0<x<1$ (since $x$ and $y$ are both nonnegative). I have the feeling that this is where I went wrong but I don't see why and how to do this correctly.

$$\int_0^1\int_{y^2}^1ye^{x^2}\mathrm{d}x\mathrm{d}y=\int_0^1\int_{0}^{\sqrt{x}}ye^{x^2}\mathrm{d}y\mathrm{d}x=\int_0^1\left[\frac{y^2}{2}e^{x^2}\right]_0^{\sqrt{x}}\mathrm{d}x=\int_0^1\frac{x}{2}e^{x^2}\mathrm{d}x$$

Now I substitute $u=x^2$, so $\mathrm{d}u=2x\mathrm{d}x$: $$\int_0^1\frac{x}{2}e^{x^2}\mathrm{d}x=\int_0^1\frac{1}{4}e^{u}\mathrm{d}u=\left[\frac{e^u}{4}\right]_0^1=\frac14(e-1).$$

Can anyone tell me what I did wrong? Thanks in advance.

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