z = (x^2/25) + (y^2/9) + 12
Now, let's set up a double integral to find the surface area of this equation in rectangular coordinates.
The surface area (S) can be calculated using the formula:
S = ∬R √[1 + (dz/dx)^2 + (dz/dy)^2] dA
In this case, the region R corresponds to the entire xy-plane.
Therefore, the double integral for the surface area in rectangular coordinates is:
S = ∬R √[1 + (dz/dx)^2 + (dz/dy)^2] dA
= ∬R √[1 + (2x/25)^2 + (2y/9)^2] dA
Since the region R is the entire xy-plane, the limits of integration for x and y are from negative infinity to positive infinity.
The double integral for the surface area in rectangular coordinates becomes:
S = ∫∫ √[1 + (2x/25)^2 + (2y/9)^2] dx dy
Please note that finding the actual numerical value of this integral would require additional information