Integral theorem (Gauss Theorem)

Question

$$\iint \vec F \cdot \hat n \, ds$$ where $\vec F= xy \vec i + y^2 \vec j + 2yz \vec k$ in the region bounded by $z=0$, $z=2$ and $x^2+y^2=4$.

I'm finding it difficult to put the correct limits of integration, especially on $dy$ and $dz$.

Answer 1

Use cylindrical coordinates for the parametrisation of the cylinder:

$\vec r(\theta, z) = (2\cos \theta, 2\sin \theta, z)$.

Then, calculate $$\frac{\delta \vec r }{\delta \theta} \times \frac{\delta \vec r }{\delta z}$$

Your integral becomes:

$$\iint_G \vec F(\vec r(\theta, z))\frac{\delta \vec r }{\delta \theta} \times \frac{\delta \vec r }{\delta z} d\theta dz$$

It is clear that to describe the cylinder, $\theta \in [0,2\pi]$, and it is also clear that $z \in [0,2]$. So these will be the bounds you can use.