How do I calculate the root mean square voltage of a square wave (with drawing)?
You have the formula right there, integration is indeed the area under the curve. Over one period, you just have to calculate the area under the voltage waveform/curve squared. It is pretty clear that the time duration between T/4 and T will be 0 even when squared. And you have,
$$V^2 * T/4$$ of area remaining. Inserting V=14 and T=6, $$14*14*6/4 = 294$$ now multiply it with 1/T. $$294*1/6 = 49$$ It is now only a matter of solving the following equation, which I'll leave as an exercise.
$$ V_{rms} = \sqrt{49} $$