Compute the flux of the vector field f⃗ =xi⃗ +yj⃗ +zk⃗ through the surface s, which is a closed cylinder of radius 1, centered on the x-axis, with −1≤x≤1, and oriented outward.

Refer to the diagram shown below.

The outward normal to the cylindrical surface is .
The given vector field is


The flux of the vector field is the surface integral


From the Gauss Divergence Theorem, the surface integral is converted into a volume integral as


Note that


Therefore