Planning of integrator and derivator circuit
I'm not sure why you have derived a range of values for R1 based on two different values of w1. Choose one value of w1 that meets the requirements i.e. 500 rad/s and calculate R1 based on C1 being 47nF.

Do the same for R2 based on (say) w2 = 20,000 rad/sec. I'd reduce the value of capacitance on this circuit to something like 10nF or 4.7nF (see small print below)

In all other respects what you have done is perfectly good. I prefer deciding on limiting values for frequency first, then doing one set of calculations.

Small print - I am assuming that you are using op-amps to create an integrator and differentiator. If you are, be aware that the integrator will require a high value feedback resistor to maintain DC conditions and stop the op-amp output clipping against either positive or negative supply rail. This high-value resistor (1Mohm minimum) will have a slight detrimental effect on it being a true integrator. For the differentiator i would also put a small resistance in series with the capacitor to prevent noise and other issues. It probably needs to be a few tens of ohms, possible even 10 ohms.

Here's a picture of a Low-pass circuit followed by a high-pass circuit - note that these only approximate to integration and differentiation: -

The blue trace is called Vint and is the output from the "integrator". The red trace is the output of both circuits together (cascaded). I have purposefully made the impedances R1 and C1 very low so that the diff circuit attached does not "load" the output too much. I've also made C2 and R2 high-ish so that they don't form a big load to the R1,C1 circuit.

I believe that if you go for 500 rad/s and 20,000 rad/s you'll need R1 = 100R, C1 = 20uF, R2 = 51K and C2 = 1nF.