dx, dy as variables, numerical integration

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

Sorry for the potentially confusing typo, (4) below should be y = r sin(T)

Thomas

Thomas wrote: