Derivation for continuity equation in integral form

In Aerodynamics there are three fundamental equations

We will study all the equations one by one. 

Let's talk about Continuity equation once.

Continuity equation is simply conservation of mass of the flowing fluid. Consider fluid flowing through the pipe. It is really not possible that fluid entering from one end of pipe vanishes while coming out of other end of the pipe(Except if its magical fluid, just kidding). This is a same thing which continuity equation tells us. That mass of flowing fluid is conserved.

Let 

'm' be the Mass of the fluid

'V' be the Volume of the fluid

'ρ' be the Density of the fluid

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As we know Density is equal to ration of mass and volume

hence 

ρ = m/V           (1)

So mass becomes,

m = ρ x V           (2)

Volume can be written as Area times thickness

i.e V = A x t         (3)

Where,

'A' is Cross section are of pipe

't' is thickness of fluid column in pipe

So Mass becomes,

m = ρ x A x t           (4)     (Replacing V by A x t)          

To find of mass flow rate, differentiation above equation with respect to time

'' be the mass flow rate

Hence

 = ρ x A x v         (5) (Differentiation of t with respect to time gives velocity of the fluid)           

Considering mass flow rate we got is for small section of fluid

So to find mass flow rate for entire fluid

We will write mass flow rate from equation (5) in integral form

Equating both the mass flow rate equations we get,

using gauss divergence theorem,

This is integral form of continuity equation