Answer
$$\frac{76}{3}-\frac{20}{3} \sqrt{5} $$
Work Step by Step
\begin{aligned}
\int_{0}^{4} \int_{0}^{5} \frac{d y d x}{\sqrt{x+y}} &=\int_{0}^{4}\left(\int_{0}^{5} \frac{d y}{\sqrt{x+y}}\right) d x \\
&=\int_{0}^{4}\left(\left.2 \sqrt{x+y}\right|_{y=0} ^{5 }\right) d x \\
&=2 \int_{0}^{4}(\sqrt{x+5}-\sqrt{x}) d x \\
&=2\left(\frac{2}{3}(x+5)^{3 / 2}-8\right)-2\left(\frac{2}{3} \cdot 5^{3 / 2}-0\right) \\
&=36-\frac{32}{3}-\frac{20}{3} \sqrt{5}\\
&=\frac{76}{3}-\frac{20}{3} \sqrt{5}
\end{aligned}
