In general if $X\subset\mathbb{P}^N$ is a smooth complete intersection of hypersurfaces of degree $d_1,...,d_c$. Then $\omega_{X}\cong\mathcal{O}_{X}(d_1+...+d_c-N-1)$. In you case $d = 5, N=4$. So $\omega_{X}\cong\mathcal{O}_{X}$ and $X$ is Calabi-Yau.

To prove this formula you have to use the adjunction formula: 

http://en.wikipedia.org/wiki/Adjunction_formula_(algebraic_geometry)

If $Y\subset X$ is a smooth subvariety of codimension one then $K_Y = (K_X+Y)_{|Y}$. If $Y$ is an hypersurface of degree $d$ in $X = \mathbb{P}^{N}$, then $K_Y = (d-n-1)H_{|Y}$ where $H$ is the hyperplane section. Here I am using $\omega_{\mathbb{P}^N}\cong\mathcal{O}_{\mathbb{P}^N}(-N-1)$. To prove the formula for complete intersections just proceed by induction.

In general any smooth hypersurface $X\subset\mathbb{P}^{N}$ of degree $N+1$ is Calabi-Yau.