Answer
$$S=\frac{3 \sqrt{3}} {5}$$
Work Step by Step
We find:
\[
\begin{array}{c}
S=\int_{0}^{1} \int_{0}^{y} \sqrt{1+2+2 y} d x d y \\
S=\iint_{R} \sqrt{\frac{\delta z}{\delta x}+\frac{\delta z}{\delta y}+\frac{\delta z}{\delta z}} d A \\
S=\int_{0}^{1} \sqrt{2 y+3}[x]_{0}^{y} d y
\end{array}
\]
\[
\int_{0}^{1} \sqrt{2 y+3} y d y=S
\]
\[
\left[\frac{1}{5}(y-1)(3+2 y)^{\frac{3}{2}}\right]=S
\]
Evaluating:
$$S=\frac{3 \sqrt{3}} {5}$$
