Given $$F(x,y,z)=(x^3+\sin(z),x^2y+\cos(z),\exp(x^2+y^2))$$ I have to evaluate the flux of $F$ through $S$, with $S$ being the surface of $Q$, such that $Q$ is bounded by the cylinder $$z=4-x^2$$ the plane $$y+z=5$$ and the planes $xy$ and $xz$.
I decided to use the Gauss theorem to evaluate it as follows:
$$\iiint_{Q}(3x^2+x^2+0)dV=\iiint_{Q}(4x^2)dV$$
My question is, may I write $Q$ as $$Q=\{(x,y,z)\mid 0\leq y\leq5-z,0\leq z\leq4-x^2,-2\leq x\leq2\}$$ because when I plotted in Geogebra, there's a gap between the cylinder and the plane $y+z=5$, so I thought, I should evaluate two integrals: one for $0\leq y\leq 1$ and another one for $1\leq y\leq5-z$.