Someone plays darts that has a join density function based on the surface z = 3 - r where z is the likelihood of the dart landing at $(r,\theta)$. How can I find the value of C so that f is a joint density function and then find the probability that the thrown dart lands in S?
$$g(r,\theta)= \begin{cases} C_{1}(3-r), & \text{if} (r,\theta)∈ R\\ 0, & \text{otherwise} \end{cases}$$
R is the region inside of r = 1 + tan $(\theta /4)$ (which includes the little loop) and S is just the region inside the inner loop.
So far this is how I am solving for C but how can I actually get the probability? Also is this correct so far?