I have the following joint probability mass function:
I have calculated a mean for R of $2/3$ and a Variance for R of $7/18$ For G, the mean is $26/21$ and the variance is $0.033$
I have this formula for the covariance: $(X+Y) = E(XY) - (E(X) \times E(Y))$
I have seen this problem solved by integration, but I have no function to integrate, only a table. How should I go about this?
We use summation, not integration. We have $$E(RG)=(1)(1)(24/84)+(1)(2)(3/84)+(2)(1)(6/84).$$ (I left out a whole bunch of $0$ terms.) Then use $\text{Cov}(R,G)=E(RG)-E(R)E(G)$.
Another way: Alternately, we find the distribution of $RG$ and then use the ordinary formula for expectation. The "possible" values of $RG$ are $0,1,2,3,4,6$. However, for computing the expectation, we do not need to know $\Pr(RG)=0$, since $RG=0$ will not contribute anything to the expectation. We also do not need to worry about "possibilities" such as $RG=3$, since the probability that $RG=3$ is $0$, so there is no contribution to the expectation.
The only interesting possibilities are $RG=1$ and $RG=2$. From the table, we have $\Pr(RG=1)=24/84$ and $\Pr(RG=2)=3/84+6/84$.