I asked this question years ago $f_X(x) \neq \frac{2 \pi \sqrt{R^2 - x^2}}{4\pi R^2}$? $X$ belongs to points uniformly distributed on the surface of a sphere., and I'm trying to use 2 other approaches, but I'm not sure if they're the right approaches.
First approach (from the joint PDF of $x, y, z$).
We know the joint PDF is the inverse of the surface area of the sphere, which is
$$ f_{XYZ}(x, y, z) = \frac{1}{4 \pi R^2}. $$
So integrating this over $y$ and $z$ with proper lower and upper limits should get us the right answer for $f_X(x)$.
Second approach
We know that the marginal pdf of $X$ is proportional to the circumference of the circle sliced from $X = x$, i.e,. a circle on the y-z plane with radius $\sqrt{R^2 - x^2}$, i.e.,
$$ f_X(x) = 2 \pi \sqrt{R^2 - x^2} $$
Are these approaches correct?