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## Main Question or Discussion Point

The mass of a sphere with density as a function of radius is

[itex]M=4\pi \int_0^r\rho(r) r^2dr[/itex]

Lets say the radius increases and decreases as a function of time t. So:

[itex]M(t)=4\pi \int_{0}^{r(t)}\rho (r) r(t)^2dr[/itex]

I want to know the basic equation describing the mass added or removed from the sphere (mass increases when radius increases, mass decreases when radius decreases) as a function of t, starting from any t. The problem is I think I must use a differential form but I'm not sure what it looks like. What then is the differential form of dM(t)/dt? I think I must use a chain rule and write:

[itex]\frac{dM}{dt}=\frac{dM}{dr}\frac{dr}{dt}[/itex]

is this right? How do I proceed to solve this with the integral?

[itex]M=4\pi \int_0^r\rho(r) r^2dr[/itex]

Lets say the radius increases and decreases as a function of time t. So:

[itex]M(t)=4\pi \int_{0}^{r(t)}\rho (r) r(t)^2dr[/itex]

I want to know the basic equation describing the mass added or removed from the sphere (mass increases when radius increases, mass decreases when radius decreases) as a function of t, starting from any t. The problem is I think I must use a differential form but I'm not sure what it looks like. What then is the differential form of dM(t)/dt? I think I must use a chain rule and write:

[itex]\frac{dM}{dt}=\frac{dM}{dr}\frac{dr}{dt}[/itex]

is this right? How do I proceed to solve this with the integral?