Flux of vector field F = xi + yj + zk across S

[schmiggy]

Homework Statement

I've attached an image with the entire question.

Homework Equations

Attached an image with relevant equations. Can't use Gauss' Divergence

The Attempt at a Solution

In the attached image I've also included the start of my calculations, I just need to see if my double integral is correct.. if it is I can easily compute it, and have done so.. however the answer I got made me doubt my working.

Naturally I would replace the x and y with rcos(theta) and rsin(theta) respectively which would become 2r^2, as cos^2(theta) + sin^2(theta) = 1.

Anyway, the answer I got using the double integral in the attached image was 384pi and can't help but feel like I'm way off. Any guidance would be greatly appreciated, thanks!

 

[I like Serena]

Homework Statement

I've attached an image with the entire question.

Homework Equations

Attached an image with relevant equations. Can't use Gauss' Divergence

The Attempt at a Solution

In the attached image I've also included the start of my calculations, I just need to see if my double integral is correct.. if it is I can easily compute it, and have done so.. however the answer I got made me doubt my working.

Naturally I would replace the x and y with rcos(theta) and rsin(theta) respectively which would become 2r^2, as cos^2(theta) + sin^2(theta) = 1.

Anyway, the answer I got using the double integral in the attached image was 384pi and can't help but feel like I'm way off. Any guidance would be greatly appreciated, thanks!

Welcome to PF, schmiggy!

Your calculations look fine, except for your upper boundary for r.
How did you get the upper boundary 4 for r?
What is the corresponding z value?

 

[schmiggy]

Hi, thanks for the reply (and the welcome!

Ahh, I think I see the problem.. at least I hope I do.. or that might be embarrassing!

Regarding the upper boundary for r, it must occur when z is at it's minimum, which in this case is 7.. therefor upper limit for r is 3.

I feel like I'm still going wrong somewhere.. like I'm missing something very basic.. I've attached my full working and hopefully you can see something I'm missing.. Thanks again!

 

[I like Serena]

Well, you're not going wrong anywhere... but you did not finish the calculation...

Apparently you did get 384pi, which does correspond to an upper boundary of r=4.

 

[schmiggy]

For some reason it looks like my image didn't get uploaded.. hopefully it works this time.. these are my current calculations.

 

[I like Serena]

All fine... except for a calculation error when you filled in r=3.

 

[schmiggy]

uggghhh... I calculated 3^3 instead of 3^4, it's late and I'm getting sloppy.. thanks for the pick up.

Final answer (369pi)/2??

 

[I like Serena]

Yep!

 

[schmiggy]

Phew! Thanks! Might take the rest of the night off I think.. if I'm missing those little errors it's time to stop!

Thanks again!