To compute the area of a surface parametrized by , you need to integrate the "surface element" (where is the cross product of vectors) over the range of and (this is a double integral). In your case, and replace and , and where and . I haven't tried much to compute the integral I obtained this way, but it doesn't look very nice anyway...
I hope you'll be able to carry on using this.
(I forgot to mention that a more convenient expression for the surface element is where , and )