(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate the surface integral [tex]\vec{F}\cdot\vec{n}\, dS[/tex]

where [itex]\vec{F}=<-y,x,0>[/itex] and S is the part of the plane [itex]z=8x-4y-5[/itex] that lies below the triangle with vertices at (0,0,0,), (0,1,0,) and (1,0,0). The orientation of S is given by theupward normal vector.answer: 2

I am not sure if I am just making a careless mistake or a conceptual one.

3. The attempt at a solution

i.) Parametrizing S gives [itex]\vec{r}(x,y)=<x, y, 8x-4y-5>[/itex]

ii.) Finding [tex]\frac{\partial r}{\partial x}\times \frac{\partial r}{\partial y}=<1,0,8>\times<0,1,-4>=-8,4,1[/tex]

iii.) Thus, [itex] \vec{F}(\vec{r}(x,y))\cdot (\frac{\partial r}{\partial x}\times \frac{\partial r}{\partial y})=<-y,x,0>\cdot<-8,4,1>=<4x+8y>[/itex]

iv) Therefore [tex]I=\int\int_D (4x+8y)\, dA[/tex]

[tex]=\int_{x=0}^1 \int_{y=0}^x (4x+8y)\,dy\, dx[/tex]

I belive that if I made an error, it was made somewhere in here and not in my integration.

Any major blunders here?

Thanks,

Casey

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# Homework Help: Surface Integral

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