I'm currently taking my third semester of calculus, and it's somewhat saddening that I don't quite understand this definition.
For single variable calculus, I get that you divide the segment between the limits of integration into pieces, and then pick a random point within each for the function value. The products of the length of the segments and the function values create Riemann rectangles that approximate the area under the curve.
What I don't understand is why the definite integral's value is the limit as the biggest segment (the norm) approaches zero. Doesn't that segment just disappear, leaving you with what you originally have, except with that specific rectangle removed? How does this better approximate the integral?
Thanks for any input.