How to use the divergence theorem to solve this question

[Leo Liu]

Homework Statement

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Relevant Equations

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The correct answer is $\frac{\pi a^2 h} 2$ by using the standard approach. However when I tried using the divergence theorem to solve this problem, I got a different answer. My work is as follows:
$$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$
$$= \iiint_D \frac{\partial y}{\partial y}\, dV$$
$$=\iiint_D \, dV=V_{cylinder}=\pi a^2h$$
Where did I go wrong?

 

[PeroK]

Is the surface integral over a closed surface?

 

[Orodruin]

Your volume is only half the cylinder.
(After you add the surfaces needed to close the surface where the integral contribution is zero.)

 

[Leo Liu]

Is the surface integral over a closed surface?

Nope. But IMO the flux passing through the led of the cylinder and the bottom of the cylinder is 0.

 

[Leo Liu]

Your volume is only half the cylinder.
(After you add the surfaces needed to close the surface where the integral contribution is zero.)

I am sorry. I don't get it.

 

[PeroK]

I am sorry. I don't get it.

You've calculated the volume of the whole cylinder. The surface is only half the cylinder.

 

[Leo Liu]

You've calculated the volume of the whole cylinder. The surface is only half the cylinder.

Okay I see -- I didn't see the condition that the surface is to the right of the xz plane. Thank you!