For the Inverse Gamma Distribution, show that
Var(Y) = Beta^2 / {[(alpha-1)^2]*[alpha-2]} , alpha > 2.
Start by computing $\displaystyle E(Y^2)$, with a density (or whatever alternative definition your teacher has choosen):
$\displaystyle f(y;\alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\al pha)}(1/y)^{\alpha} exp(-\beta/y)$
You will not need to do the integral, as with a bit of jiggery-pokery you should be able to rearrange the integrand to be a multiple of the density of another Inverse Gamma distribution.
CB
It hardly matters, with whatever definition the OP has of the density (or rather of what the parameters are) the basic method will work, and as we are not going to do it for them there is no problem in presenting the method for one version, it will work with the other/s though with a different final answer. It just gives them an additional opportunity to contribute to the solution themselves.
CB