The above equation can be re-written as follows:
$\sqrt{x} + \sqrt{y} = \sqrt{6}$
This depicts the graphical notation of the above curve:
The curve can be re-written as follows:
$\sqrt{y} = \sqrt{6} - \sqrt{x}$
$y = 6 + x + \left ( 2\ast \sqrt{6}\ast \sqrt{x}\right )$
The value of x varies from x = 0 to x = 6
Area under the curve = $\int_{0}^{6}\left [6 + x + \left ( 2\ast \sqrt{6}\ast \sqrt{x}\right )\right ]dx$
= $\left [ 6\ast x + {\frac{x^{2}}{2}} + 3\ast\sqrt{6}\ast x^{\frac{3}{2}}\right ]$
On computing this, you get Area = 6.
Thus, answer is option (C).