I have a function, f(x, y)), integrated over an area defined as a

quarter circle in quadrant 1. That is, an area defined between 0 and 90

degrees with a radius of 1. I can define this result in Cartesian

coordinates as (using "S" as the integral sign):

(1) SS f(x, y) dx dy

or, in polar coordinates as,

(2) SS f(r) r dr dT

where dT is shorthand for d(Theta).

I can make the conversion from Cartesian to polar using the

transformations of (3) x = r cos(T) and (4) y = sin(T). dx and dy just

fall out of those relationships.

My question comes thus. Can I simply treat dx and dy as any other

variable, multiply, simplify them and do the integration? I would have

said yes. However, in a real example where,

(5) f(x, y) = square root (1 - x2 - y2) = f(r) = square root (1 - r2)

I did a numerical integration. I get the expected answer, pi / 6, by

doing (1) or (2). But, if I expand (1) using (3) and (4) the numerical

integration does not work. Any thoughts?

Best regards,

Thomas