<s>[INST] Integration of double integral Integration of double integral of.  
\begin{equation}
\int_{-\frac{1}{4}}^{0}{\int_{\frac{1}{2}-\sqrt{x+\frac{1}{4}}}^{\frac{1}{2}+\sqrt{x+\frac{1}{4}}}} {e^{y^2}dy  dx} + \int_{0}^{2}{\int_{-1+\sqrt{x+1}}^{1/2+\sqrt{x+\frac{1}{4}}}} {e^{y^2}dy  dx} + \int_{2}^{8}{\int_{-1+\sqrt{x+1}}^{2}} {e^{y^2}dy  dx} 
\end{equation}
 [/INST] If we reverse the order of integration the integrands can be transformed into $ye^{y^2}\,dy$ and $y^2e^{y^2}\,dy$.
Here is the region of integration from desmos.com/calculator
First divide the region into five sections having unique left and right bounding functions of $y$

Giving the following region and subregions

Then change the order of integration over the five sub-regions to obtain
\begin{eqnarray}
& &\int_0^1\int_{y^2-y}^{0}e^{y^2}dxdy\\&+&\int_{0}^{\sqrt{3}-1}\int_{0}^{y^2+2y}e^{y^2}dxdy\\
&+&2(2-\sqrt{3})\\
&+&\int_{1}^{2}\int_{y^2-y}^2e^{y^2}dxdy\\
&+&\int_{\sqrt{3}-1}^{2}\int_2^{y^2+2y}e^{y^2}dxdy
\end{eqnarray}
You should double check the limits but I believe this is correct.
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