Answer
$$123.8667$$
Work Step by Step
\begin{aligned}
\int_{0}^{4} \int_{0}^{9} \sqrt{x+4 y} d x d y &=\left.\int_{0}^{4} \frac{2}{3}(x+4 y)^{3 / 2}\right|_{x=0} ^{9} d y\\
&=\int_{0}^{4} \frac{2}{3}\left((9+4 y)^{3 / 2}-(4 y)^{3 / 2}\right) d y \\
&=\left.\frac{2}{3}\left(\frac{2}{5 \cdot 4}(9+4 y)^{5 / 2}-\frac{2}{5 \cdot 4}(4 y)^{5 / 2}\right)\right|_{0} ^{4} \\
&=\frac{1}{15}\left(5^{5}-4^{5}\right)-\frac{1}{15}\left(3^{5}-0\right) \approx 123.8667
\end{aligned}
