It seems that you are doing it in terms of y. This is the area of a quarter of a circle in the first quadrant, so for y, the value should go from 0 to √10.
We also have to express the function in terms of x:
x^2+y^2=10, x^2=10-y^2 so f(y)=√(10-y^2)
The area using definite integral, then, should be A = (lim n->∞) (√10 /n) Σ (i=0~n) f(<√10>/n * i).
I hope I'm right...?