navier-stokes order of magnitude

hey pf!

i have a quick question with order of magnitudes, specifically relating to navier-stokes equations. when considering the pressure term of the x-momentum we have that ##-\frac{\partial P}{\partial x}## is of order ##\mathcal{O}\big(\frac{U^2}{L}\big)## and i know ##-\frac{\partial P}{\partial y}## is of order ##\mathcal{O}\big(\frac{U^2}{L} \frac{\delta}{L}\big)##. i should say that i am inside a boundary layer, and that ##L## is the characteristic length scale parallel to ##x##, and that ##\delta## is the characteristic length scale parallel to ##y##. we are only in a 2-D flow over a flat plate (although i'm not sure this info is necessary).

my question is, if ##-\frac{\partial P}{\partial x}## is of order ##\mathcal{O}\big(\frac{U^2}{L}\big)## then if we seek order of ##P## couldn't we simply take ##\int -\frac{\partial P}{\partial x} dx## which seems to give us order ##\mathcal{O}\big(\frac{U^2}{L}{L}\big)##. now that we have ##P## if we wanted ##\frac{\partial P}{\partial y}## then take ##\frac{\partial}{\partial y} \int -\frac{\partial P}{\partial x} dx## which gives us order ##\mathcal{O}\big(\frac{U^2}{L}\frac{L}{\delta}\big)##. yet i already know this result is incorrect (in fact, the reciprocal of what should happen!). can anyone offer any insight here?

thanks!