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Intro. to Integrals

  1. May 24, 2009 #1
    I`m reading on Integrals, and at the introductions author mentions three basic concepts, The Rectangle Property, The Addition Property, and The Comparison Property.

    I understand what the 1st and 3rd properties mean, and I have a question concerning the 2nd.

    "The Adittion Property: The area of a region composed of several smaller regions that overlap in at most a line segment is the sum of areas of the smaller regions."

    I don`t understand what that part in red means??
    I don`t understand what "overlap" means, nor do I understand what "in at most a line segment" means?

    Thanx in advance
     
  2. jcsd
  3. May 24, 2009 #2

    HallsofIvy

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    I'm sure you could look up "overlap" in a dictionary! The two rectangles with boundaries [itex]0\le x\le 6[/itex], [itex]0\le y\le 1[/itex] and [itex]5\le x\le 7[/itex], [itex]0\le y\le 1[/itex] "overlap" on the rectangle bounded by x= 5, x= 6, y= 0 and y= 1- that region is in both rectangles.

    The two rectangles [itex]0\le x\le 6[/itex], [itex]0\le y\le 1[/itex] and [itex]6\le x\le 7[/itex], [itex]0\le y\le 1[/itex] "overlap" only on the line x= 6- they have only that line segment in common.

    Finally, the two rectangles [itex]0\le x\le 6[/itex], [itex]0\le y\le 1[/itex], and [itex]7\le x\le 8[/itex], [itex]0\le y\le 1[/itex] do not overlap at all- they have no points in common.
     
  4. May 24, 2009 #3

    tiny-tim

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    Hi wajed! :smile:

    (btw, it looks better if you type ' rather than ` in "don't" etc … the ` takes up too much room! :wink:)
    And a line segment is just part of a line …

    "segment" from a Latin word meaning to cut …

    so [0,1] is a segment of the real line.

    In other words, "in at most a line segment" means (in this context) zero area. :wink:
     
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