Integration boundaries for \(X+Y\leq 6\):

X: 1 to 6-y

Y: 2 to 4

Integration:

\(P(X+Y\leq 6)=\int_{y=2}^{4}\int_{x=1}^{6-y}0.0186x^{2}e^{\frac{-y}{3}}dxdy\)

\(=0.0186\int_{y=2}^{4}e^{\frac{-y}{3}}[\frac{x^{3}}{3}]_{1}^{6-y}dy\)

\(=0.0186\int_{2}^{4}e^{\frac{-y}{3}}[\frac{(6-y)^{3}}{3}-\frac{1}{3}]dy\)

\(=0.0062\int_{2}^{4}e^{\frac{-y}{3}}[(6-y)^{3}-1]dy\)

\(=0.0062\cdot24.0489\)

=0.1491