Is this the proper way I'm calculating RMS?
You can't just compute the RMS values of each region separately. Imagine if the second region of the graph was zero instead of 200. By your method, the RMS of the total would be the same regardless of the length of time the second region was at zero - which is clearly wrong.

You need to compute the area of the entire graph (squared) first, then take the mean , then take the square root :-

$$\sqrt{\frac{1}{10}\Big(\int_0^51600x^2\,\mathrm dx + \int_5^{10}40000\,\mathrm dx\Big)}$$