Do you realize this comment of yours is worth more than years of studying (high school + Bc.)? Perhaps schools are better at your location but the ones I've attended never explained this, just write that down, this equals that, memorize and f-ck you. Thanks G-d we have the Internet and people like you nowadays...
I think that Knuth's concrete mathematics made this connection at some point - the difference between the infinitely small abstraction of an integral and a big-sigma sum as being discreet quantities that can be measured by actual numbers that measure a thing.
I remember when I had calculus in school and the textbook was a huge book weighing like five pounds...and my mom's calculus textbook from the 1950s was this tiny little thin book. But they hadn't added anything to the subject of calculus since then!
