Find the flux of F through S ?

[CalleighMay]

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help!

I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesday he asks us to discuss the problems as a class, seeing which ones of us know our stuff =P

Basically, i want to ask you guys what you think about these problems as i do them along before i have my discussion. I really want to make a lasting impression on my professor by "knowing my stuff" -to show him i can do it! All's i need is a little help! Would you guys mind giving me some help?

We are using the textbook Calculus 8th edition by Larson, Hostetler and Edwards and the problems come from the book.

The problem is on pg 1118 in chapter 15.6 in the text, number 24. It reads:

Find the flux of F through S
It gives:
Integral (with S at bottom) of the integral of F (with a dot) N dS
where N is the upward normal vector to S
then it gives for this specific problem,
F(x,y,x)=xi+yj
and
S: 2x+3y+z=6, first octant

I'll be honest and say that i have no idea what's going on here. I looked at the other problems in this problem set and still have no idea what to do... :(

Any help would be greatly appreciated. Thanks guys :(

 

[Defennder]

The general method for doing such problems is to first parametrise the surface over which the integral is to be evaluated. Then express F in terms of the parametric representation of the surface, finally evaluating the dot product of the normal vector of the differential surface as a double integral with appropriate methods. It's hard to explain what to do here; you need to go read the relevant sections in the textbook.

 

[dynamicsolo]

The problem is describing the way in which the vector field, given by F(x,y,x)=xi+yj , is passing through the tilted plane 2x+3y+z=6 , which is quantified by what is called the flux through the surface S.

To start this problem, you should know how to find the dot product of two vectors, one of which is F and the other is the normal (perpendicular) vector to the tilted plane. (We have the coefficients in the equation for that plane, so the components of the normal vector are available.)

Now, if that dot product were a constant, we could do this problem without any calculus at all. Unfortunately, the dot product here is a function of x and y . It must be integrated over the "footprint" on the xy-plane that the tilted plane covers in the first octant, where x, y, and z are all positive. What is the shape of that region of the xy-plane? The dot product function would need to be integrated over that region. In this situation, it is not so easy to avoid having to do a double integral (unlike your centroid problem)...