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?