Sorry if this sounds like a dumb question, but why is the effective value of a sine wave 0.707, as opposed to 0.637 which is the value generated by finding the definite integral over the domain [0,∏] divided length of the domain?
RMS means "root mean SQUARED", so we have to square the sine wave before integrating. Regarding why we use this measurement, it is essentially a generalization of Euclidean distance: ##\sqrt{\int |x(t)|^2 dt}## is a limiting form of ##\sqrt{|x(t_1)|^2 + |x(t_2)|^2 + \ldots + |x(t_n)|^2}##, which is the distance between the point ##(x(t_1), x(t_2), \ldots, x(t_n))## and the origin. There are many other reasons to prefer the RMS as well: it plays nicely with how we measure the energy in a random quantity (variable or process), namely the standard deviation. Also, the RMS of a function/signal is preserved when we transform to the frequency domain via the Fourier transform. Mathematically, "RMS" is also a common way to measure the norm ("size") of a function: we call it the ##L^2## norm. Working in the ##L^2## space is very nice because it is a Hilbert space, unlike the other ##L^p## spaces, and because the Fourier transform is an isometry on the ##L^2## space. Don't worry if these terms are unfamiliar - you may see them eventually if you study advanced mathematics or physics, but otherwise you can probably live a perfectly happy life if you never hear about them again.
By the way, the calculation you performed is also a common way of measuring the size of a function/signal. In mathematics we call it the ##L^1## norm: ##\int |x(t)| dt##. It is a limiting form of ##|x(t_1)| + |x(t_2)| + \ldots + |x(t_n)|##, which is another way of measuring the distance between a point and the origin, assuming you are constrained to travel along an orthogonal "grid" to get to the point.
The 0.707 figure is relevant where we are concerned with heating, or the heat produced by that waveform. So a sinewave of amplitude A_{v} produces heat in a resistance equivalent to that produced by DC of amplitude 0.707A_{v}, since instantaneous power = i^{2}(t).R The 0.637 figure also has its uses, but to situations where we are concerned with average. For example, an electromagnet is roughly linear, so if you applied a rectified sinewave of amplitude A_{v} to the windings of an electromagnet, the field strength produced will have an average value equal to that produced by applying DC of magnitude 0.637A_{v} to the windings.