Question

Evaluate the flux integral

FdS
where F=
(1y,5z,5x) and is the surface of the plane 2x+4y+z=8 in the first octant oriented upward.

Answer 1

i remember something about F.n

Answer 2

or to be more vague\[\int\vec F\dot{}\frac{\nabla f}{|\nabla f|}\sqrt{something??}\]

Answer 3

any of this ring a bell?

Answer 4

for a refresher, try this:

http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx

Answer 5

Have you tried anything? I can explain it but I'm not going to do it for you.

Answer 6

is the integral \[\int\limits_{?}^{?} (78y+45x-160) \] but idk the bounds.

Answer 7

Let:
\[\vec{\Phi}(x,y)=(x,y,8-2x-4y)\]
Find:
\[\frac{\partial \vec{\Phi}}{\partial x} \times \frac{\partial \vec{\Phi}}{\partial y}\]
Once you have that you take the dot product with:
\[\vec{F}(\vec{\Phi}(x,y))\]
From your cross product you can determine if its orientation preserving or reversing. And if the plane is bounded in the first octant then:
\[z=0 \implies y=\frac{-x}{2}+2 \implies 0\le y \le \frac{-x}{2}+2 \implies 0 \le x \le 4\]
I believe because you're looking at the "shadow" of the plane in the x-y plane and you have something that looks like:


I believe.