Hi,

it looks like your doubt is not related to the bivariate but rather to the definition of the variance?

Once you marginalize the joint PDF respect the r.v. X you obtain your univariate PDF g(x), from which you can compute the mean value of the r.v. X and then just compute the variance (see Variance - Wikipedia, the free encyclopedia).

The fact that equals is by definition of the expectation operator (Expected value - Wikipedia, the free encyclopedia - see the part related to E{g(x)}).