I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post).
They ask me to find the mass flow rate passing through a surface, where the velocity field is given by $\vec{V}(x,y,z) = (x, -4y, z)$, the density is $\delta (x,y,z)=x^2 + z^2$ and the surface is defined as:
$S: {\frac{x^2}{4}+y^2=1, 0 \leq z \leq 2-x}$
I know I could use the divergence theorem, but they ask me to do this with and without it, and I'm having troubles trying to do it with the theorem.
First, to find the mass flow rate passing through a surface S, I think you have to calculate: $\int_{S}^{}\int_{}^{} \delta \vec{V} d\vec{S}$.
Second, the surface $S$ is an elliptical cylinder, where $0 \leq z \leq 2-x$, so it's not a closed surface.
Considering the above:
$\int_{}^{}\int_{S}^{} \delta \vec{V} d\vec{S} = \int_{}^{}\int_{}^{}\int_{V}^{} \nabla \delta \vec{V} dV - \int_{}^{}\int_{t_1}^{} \delta \vec{V} d\vec{S} - \int_{}^{}\int_{t_2}^{} \delta \vec{V} d\vec{S}$, where $t_1$ and $t_2$ are the necessary surfaces so that the surface $S \cup t_1 \cup t_2$ is closed.
Now, according to my calculations $\nabla \delta \vec{V} = 0$, so we need to calculate: $\int_{}^{}\int_{t_1}^{} \delta \vec{V} d\vec{S}$ and $\int_{}^{}\int_{t_2}^{} \delta \vec{V} d\vec{S}$
Let's start with $\int_{}^{}\int_{t_1}^{} \delta \vec{V} d\vec{S}$, $t_1$ will represent the cover of the cylinder, where $z = 2-x$, so we can parameterize t1 as: $\vec{r} = (x,y,2-x)$ and then $d\vec{S} = (1,0,1)$, $\vec{V}(\vec{r}) = (x, -4y, 2-x)$, so:
\begin{align*} \int_{}^{}\int_{t_1}^{} \delta \vec{V} d\vec{S} = &\int_{}^{}\int_{t_1}^{} (x^2 + z^2)(x, -4y, 2-x)(1,0,1)dA \\ &\int_{}^{}\int_{t_1}^{} 2(x^2+z^2)dA \end{align*}
My first question is, up to this step, am I doing everything right?
Second, how do I choose the limits of integration for the last integral?