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The equation $x^2 + y^2 = 9$ describes a full circle centered at the origin with radius 3, which includes both upper and lower branches. However, the original function is defined using the principal square root, $y = \sqrt{9-x^2}$.

Conditions: Working over the real numbers.; Using the principal (nonnegative) square root convention.; The underlying relation is the circle $x^2 + y^2 = 9$.

The calculation averages complex points sampled uniformly in *time*, not uniformly along the *arc length* of the trajectory. Because the signal $g(t)$ modulates the radius and the rotation speed varies with frequency, equal time intervals do not correspond to equal distances traveled along the curve.

Conditions: Sampling is done at uniform time intervals $t_k$.; The path is defined by $z_f(t) = g(t)e^{-2\pi i ft}$.