Flux integral over a parabolic cylinder

[Forcefedglas]

Homework Statement
Evaluate $\int\int_S \textbf{F}\cdot\textbf{n} dS$ where $\textbf{F}=(z^2-x)\textbf{i}-xy\textbf{j}+3z\textbf{k}$ and S is the surface region bounded by $z = 4-y^2, x=0, x=3$ and the x-y plane with $\textbf{n}$ directed outward to S.

The attempt at a solution

I've worked out the correct answer but can't seem to fully understand why that is. I tried splitting up the flux integral into 3 separate surfaces: 1 for the parabola at x=3, another for the parabola at x=0, and lastly a parametric surface between them. At each parabola I just evaluated the flux integral in cartesian coordinates, which were $\int_{-2}^2 \int_0^{4-y^2}(2y\textbf{j}+\textbf{k})\cdot((z^2-x)\textbf{i}-xy\textbf{j}+3z\textbf{k})$, which worked out to be 256/5 and 0, at x=0 and x=3 respectively.

I parameterized the parabolic cylinder as $\mu\textbf{i}+\lambda\textbf{j}+(4-\lambda^2)\textbf{k}$, so the flux integral for this was $\int_0^2 \int_0^3(2\lambda\textbf{j}+\textbf{k})\cdot((z^2-x)\textbf{i}-\mu\lambda\textbf{j}+(12-3\lambda^2)\textbf{k})d\mu d\lambda$ which works out to be 48, which is the correct answer. This might seem like a dumb question but I've been staring at it for hours and can't understand why the value of the flux integral at the x=0 parabola is ignored. I considered the possibility that it excludes the surfaces at x=0 and x=3 but similarly worded questions did not do this. Any help/tips will be appreciated, thanks!

 

[vela]

Homework Statement
Evaluate $\int\int_S \textbf{F}\cdot\textbf{n} dS$ where $\textbf{F}=(z^2-x)\textbf{i}-xy\textbf{j}+3z\textbf{k}$ and S is the surface region bounded by $z = 4-y^2, x=0, x=3$ and the x-y plane with $\textbf{n}$ directed outward to S.

The attempt at a solution

I've worked out the correct answer but can't seem to fully understand why that is. I tried splitting up the flux integral into 3 separate surfaces: 1 for the parabola at x=3, another for the parabola at x=0, and lastly a parametric surface between them. At each parabola I just evaluated the flux integral in cartesian coordinates, which were $\int_{-2}^2 \int_0^{4-y^2}(2y\textbf{j}+\textbf{k})\cdot((z-x^2)\textbf{i}-xy\textbf{j}+3z\textbf{k})$, which worked out to be 256/5 and 0, at x=0 and x=3 respectively.

First, is the x-component of the vector field $z^2-x$ or $z-x^2$? You used the wrong normal for evaluating the surface integrals on the x=0 and x=3 planes.

I parameterized the parabolic cylinder as $\int_0^2\int_0^3\mu\textbf{i}+\lambda\textbf{j}+(4-\lambda^2)\textbf{k}$, so the flux integral for this was $(2\lambda\textbf{j}+\textbf{k})\cdot((z-x^2)\textbf{i}-\mu\lambda\textbf{j}+(12-3\lambda^2)\textbf{k})d\mu d\lambda$ which works out to be 48, which is the correct answer. This might seem like a dumb question but I've been staring at it for hours and can't understand why the value of the flux integral at the x=0 parabola is ignored. I considered the possibility that it excludes the surfaces at x=0 and x=3 but similarly worded questions did not do this. Any help/tips will be appreciated, thanks!

There seems to be numerous errors or typos in what you've written here. Could you please clean it up?

 

[Forcefedglas]

First, is the x-component of the vector field $z^2-x$ or $z-x^2$? You used the wrong normal for evaluating the surface integrals on the x=0 and x=3 planes.There seems to be numerous errors or typos in what you've written here. Could you please clean it up?

I fixed up most of the errors I think (left i component in terms of z and x since it goes to 0 anyway). The x component was supposed to be $z^2-x$. Would the correct normals in the x=0 and x=3 planes be the unit vector i at x=3 and -i at x=0?

 

[vela]

I think you meant for the limits for $y$ on the last integral to be $-2$ and $2$. Yes, the normals are $\pm \hat i$. I get the same result you do integrating over just the curved part of the surface. With the three flat surface included, I get 16 for the total flux.

 

[Forcefedglas]

I think you meant for the limits for $y$ on the last integral to be $-2$ and $2$. Yes, the normals are $\pm \hat i$. I get the same result you do integrating over just the curved part of the surface. With the three flat surface included, I get 16 for the total flux.

Guess the given answer must be off then, thanks for clearing that up.