
Finding a Constant
Hello, One of my homework questions ask me to find the value of c where the joint density function of x,y = c*g(x^2+y^2) where g is nonnegative and the integral of g(t) from 0 to infinity converges. How would I go about answering this question if I do not know what the function g is? Any help is much appreciated.
Thank You

The double integral must equal one.
You can switch to polar and say that
$\displaystyle 1=c\int_0^{2\pi}\int_0^{\infty} g(r^2)rdrd\theta $
If g is only a function of r then...let
$\displaystyle t=r^2$ and $\displaystyle dt=2rdr$
$\displaystyle 1=c\pi\int_0^{\infty} g(t)dt$
so $\displaystyle c={1\over \pi\int_0^{\infty} g(t)dt}$