integrating m/s with a double integral

Question

If I double integrate $f(x,y)$ with unit $m/s$, will the units become $m, m^2/s$ or $m/s^2$? I know the derivative of $m/s$ is just $m$.

Answer 1

If I well understand you have a function $f(x,y)$ the has the dimension of a velocity $(m/s)$. The dimension of the integral depends from the dimensions of $x$ and $y$. ù

As example, if $x$ has the dimension of space $(m)$ and $y$ has dimension of time $(s)$, when you integrate the terms in the integral is the product of the function and two differential, and this product gives the dimension of the integrated quantity, as in the case:

$ f(x,y)dxdy $

but, if $x,y$ are length $(m)$ than $f(x,y)dxdy$ has dimension $(m/s)(m)(m)$

that has dimension $(m/s)(m)(s)=(m^2)= (m^3/s)$