Can Mass be Found Using Surface Integral and Density?

[chetzread]

in part b , we can find mass by density x area ?
is it because of the thin plate, so, the thickness of plate can be ignored?

 

[BvU]

There is no ignoring the thickness going on. The density function is a function of only two variables (*), so it provides mass/area, not mass/volume.

(*) as pointed out with the z=f(x,y) callout

 

[chetzread]

There is no ignoring the thickness going on. The density function is a function of only two variables (*), so it provides mass/area, not mass/volume.

There is no ignoring the thickness going on. The density function is a function of only two variables (*), so it provides mass/area, not mass/volume.

(*) as pointed out with the z=f(x,y) callout

(*) as pointed out with the z=f(x,y) callout

There is no ignoring the thickness going on. The density function is a function of only two variables (*), so it provides mass/area, not mass/volume.

(*) as pointed out with the z=f(x,y) callout

There is no ignoring the thickness going on. The density function is a function of only two variables (*), so it provides mass/area, not mass/volume.

(*) as pointed out with the z=f(x,y) callout

so, z=f(x,y) provide info that density depends on 2 variables only?

 

[BvU]

Yes $$\rho(x,y,z) = \rho(x,y,3-x-y) = \rho(x,y) $$it is multiplied with something of dimension length² so $\rho$ has the dimension mass/area