Doubt about a procedure

[VGNDISCPLN]

Hi everyone i am currently studying transport phenomena by myself using a book called introductory transport phenomena, by trying to understand one equation i do not realize what tricks is the author using, and if i try to make it step by step i don´t get the same result has he´s. Here is the equation.

Where rho, g, delta, beta are all constants. I dont understand why he take out the integral with respect to dy and W. Could someone make a development step by step including all asumptions to get this result. I will apprecieate so much for your help.

 

[HallsofIvy]

First there is no integral with respect to W! You mean the integral with respect to y that has W as an upper bound. It can be "taken out" because the integrand is \(\displaystyle v_z(x)\) which is a function of x only.

\(\displaystyle \int_0^W\int_0^\delta v_z(x)dxdy=\left(\int_0^Wdy\right)\left(\int_0^\delta v_z(x) dx\right)\)

And since the integral with respect to y has "1" as integrand \(\displaystyle \int_0^W dy= \left[y\right]_0^W= W- 0= W\).

 

[VGNDISCPLN]

Thank you Hallsofvy, you are right, my mind got blocked.