Fluids Mass Balance Integral Question | Homework Statement Explained

[Saladsamurai]

Homework Statement

I have include the schematic for visual purposes only. My question is about an integral, so a knowledge of Fluids is not necessarily required to answer my question.

Homework Equations

This is the mass balance in a form that is suitable for this problem:

$$\frac{d}{d\,t}(\int_{vol}\rho\,dV) + \int_{surf}\rho(\mathbf{v}\cdot\mathbf{n})d\,A = 0$$

Now the integral that I am concerned with is this next line of the above:

$$\frac{d}{d\,t}(\int_{vol}\rho A_b\,dh) + \int{surf}\rho\sqrt{2gh}\,dA = 0$$

Now, in the solution, in the 2nd term, they 'pulled' $\sqrt{2gh}$ through the integral.

I know that 'h' varies with 't' but it does not vary with 'A.' This is why they did that right?

I am just trying to see why that was legal, but I think that I get it. Just another case of
me 'thinking out loud' again.

 

[Redbelly98]

Now, in the solution, in the 2nd term, they 'pulled' $\sqrt{2gh}$ through the integral.

I know that 'h' varies with 't' but it does not vary with 'A.' This is why they did that right?

Yes, you got it.

 

[Saladsamurai]

Thank you!