Which expression is the general definition of magnetic flux through a surface?

A rectangular surface lies in the $xy$-plane, has length $L$ in the $y$-direction, and is partitioned into narrow strips of width $dx$ along $x$. If the chosen area vector points in the $+z$ direction, which differential area vector is correct?

A rectangular loop of width $4$ $\text{m}$ in the $y$-direction extends from $x=1$ $\text{m}$ to $x=3$ $\text{m}$. The magnetic field is $B\left(x\right)=2x$ $\text{T}$ in the $+z$ direction, and the area vector is also $+z$. Which flux integral is correct?

A rectangular loop extends from $x=0$ to $x=1.0$ $\text{m}$ and has height $3.0$ $\text{m}$. The magnetic field is $B\left(x\right)=(5+2x)$ $\text{T}$ in the $+z$ direction, and the area vector is also $+z$. What is the flux through the loop?

A flat surface of area $4.0$ ${\text{m}}^2$ is in a magnetic field $\overrightarrow{B}=(2\hat{i}+3\hat{k})$ $\text{T}$. The surface area vector is $\overrightarrow{A}=4.0\hat{k}$ ${\text{m}}^2$. What is the magnetic flux?