So what is the question? Are you

1. integrating 5/sqrt(25-x^2-y^2) over the area of the portion of the sphere x^2+y^2+z^2=25 that is above the region in the first quadrant bounded by the graphs of x=0, y=0, 4x^2+y^2=25?

Or are you just wanting to

2. find the surface area of the portion of the sphere x^2+y^2+z^2=25 that is above the region in the first quadrant bounded by the graphs of x=0, y=0, 4x^2+y^2=25?