Apostol page 429, problem 4

Is there a better way to set up this problem or have I made a mistake along the way?
(ie easier to integrate by different parameterization)

1. The problem statement, all variables and given/known data
Find the surface area of the surface z^2=2xy lying above the xy plane and bounded by x=2 and y=1.

2. Relevant equations
<br />
S=r(T)<br />
=\bigg(<br />
X(x,y),Y(x,y),Z(x,y)<br />
\bigg)<br />
=\bigg(<br />
x,y,\sqrt{2xy}<br />
\bigg)<br />
<br />
\frac{\partial r}{\partial x}=(1,0,\frac{\sqrt{2y}}{2\sqrt{x}})<br />
<br />
\frac{\partial r}{\partial y}=(0,1,\frac{\sqrt{2x}}{2\sqrt{y}})<br />
<br />
\frac{\partial r}{\partial x}\times\frac{\partial r}{\partial y}<br />
=\bigg(<br />
-\frac{\sqrt{2y}}{2\sqrt{x}},-\frac{\sqrt{2x}}{2\sqrt{y}},1<br />
\bigg)<br />
<br />
\left|\left|\frac{\partial r}{\partial x}\times\frac{\partial r}{\partial t}\right|\right|<br />
=\sqrt{1+\frac{2y}{4x}+\frac{2x}{4y}}<br />
<br />
a(S)=\int_0^1 \int_0^2 \sqrt{1+\frac{2y}{4x}+\frac{2x}{4y}}\,dx\,dy<br />