Flux Calculation for Radial Vector Field through Domain Boundary

[s3a]

Homework Statement

Find the outward flux of the radial vector field F(x,y,z) = x i^ + y j^ + z k^ through the boundary of domain in R^3 given by two inequalities x^2 + y^2 + z^2 ≤ 2 and z ≥ x^2 + y^2.

Homework Equations

Divergence theorem: ∫∫_S F ⋅ n^ = ∫∫∫_D div F dV

The Attempt at a Solution

Is the final answer correct in TheSolution.jpg (because I get 2*pi/3 * (2^(3/2) - 1) - pi/2)?

If someone could check if I'm right or wrong, I would REALLY appreciate it!

 

[s3a]

Actually, I get π * [2(2^[3/2] - 1) - 3/2] (where my answer's 2^[3/2] differs with the √(2) in the attachment).

 

[STEMucator]

There appears to be a typo near the front of the answer. That $2$ should not be in front of the divergence integral.

 

[s3a]

Thanks for pointing that out.

Are the parts where the author begins to use the letter "t" and, most importantly, the final answer correct?

 

[STEMucator]

Wolfram gives $V(D) = \frac{\pi}{6}(8 \sqrt{2} - 7) = 2.26$ : http://www.wolframalpha.com/input/?...u005f0&f8=1&f=DoubleIntegral.rangeend2\u005f1

Your answer appears to be wrong.

 

[s3a]

Actually, my answer (from the second post in this thread) is the same as three times yours, so it appears that I was correct and that the solution was incorrect.

I'm posting my latest work, just in case I'm right by fluke (so please confirm).

 

[STEMucator]

Sorry about that. I forgot the 3 so the answer should be 6.78.

 

[s3a]

That's the same thing I got (and not what the solution that I attached in my first post in this thread got).

It must not have seemed like a lot to you, but your confirmation was very helpful to me, so thanks for helping me confirm! :)

 

[STEMucator]

Wolfram is good for self error checking, but matlab's dblquad and triplequad functions are also a very good option.

 

[s3a]

Thanks as well for mentioning those, but I meant having another human being to confirm was useful, because I'm stressed and exhausted and studying for exams, so I wanted to make sure I wasn't making a mistake in the setup of the computation (because one makes more mistakes when stressed and/or exhausted).