Integral coordinates are coordinates that are whole numbers. Integral coordinates cannot be fractional or have decimals. They are used, generally, for gross designations.
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In reimann stieltjes integral if we assume a(x) = x then it becomes reimann integral so we can say R-S integral is generalized form of reimann integral.
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Particular integral is finding what the integral is for example the integral of 2x is x^2 + C. Finding the particular solution would be finding what C equals from the particular integral.
equinoctial system of coordinates
It's not. It depends on the method you use for summation whether summation > integral or integral > summation.