How do I take z in terms of x and y? F=yI+2J+xzK m/sec; show flux of water is 69 m^3/

[Abhishekdas]

The question:

If velocity vector is F = y I + 2 J + xz K m/sec, show that the flux of water through the parabolic cylinder y = x², 0 < x < 3, 0 < z < 2 is 69 m³/sec.

What I've tried:


I am not understanding which region I should consider and what should I do with the z term. Any hint or help would be really appreciated. Thanks in advance for your precious time.

 

[Dr.Peterson]

The question:
View attachment 10497
What I've tried:
View attachment 10498

I am not understanding which region I should consider and what should I do with the z term. Any hint or help would be really appreciated. Thanks in advance for your precious time.

I think the problem may be that you are picturing the wrong region. It doesn't lie in the xy plane, so you won't be integrating over x and y. Take your drawing and add in a z axis pointing toward you; then take the arc of the parabola from (0,0) to (3,9), and move it from the z=0 plane to the z=2 plane. The points through which the arc moves form a cylinder, in the generic sense, in this case like a bent rectangle. It looks something like this:

You should have learned how to integrate over a surface like this; you will probably want to integrate with respect to x and z. You'll need to know the normal vector to this surface.

 

[Vedantjdhv]

Can i get the whole solution

 

[stapel]

Can i get the whole solution

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