Alan Smithee has kindly offered his calculus notes for distribution freely on this web site. A preliminary version is available here for viewing (find the link at bottom of this page).
The Alan Smithee lecture notes give an account of the natural integral on the real line, suitable for classroom presentations. There are other accounts of this integral but they are more elaborate and would require rather more dedication to this theory. (They also are more expensive.)
The notes are hyperlinked making it convenient to read by computer. E.g., proofs are given in a later chapter but hyperlinks enable the reader to go back and forth easily.