Am I doing this gamma distribution correctly?
Calculate $P(Y>4)$ while $Y\sim \Gamma(a,b) \text{ with } a = 2, b =3$
$P(Y > 4) = 1 - P(Y \leq 4)$
with pdf $f(y)$ given
$$f(y) = \frac{1}{(b^a\Gamma(a))}*y^{a-1}*\exp(\frac{-y}{b})=\frac{1}{9}y\exp(\frac{-y}{3}) $$ therefore we get $$ P(Y>4)=1 - P(Y \leq 4)= 1 - \int_{0}^{4}\frac{1}{9}y\exp(\frac{-y}{3})dy\approx0.615 $$
yes your result is correct, since
$$ P(Y>4)=1-P(Y\leq 4)=1- F_{gamma}(4,2,3)\approx 0.615 \text{ with }\Gamma(2)=1 $$ I just want to point out that there are different ways of defining the parameter $a,b$ of the corresponding Gamma distribution. So just make sure you took the right pdf and you're fine.
bests