Intro. to Integrals

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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
 

Answers and Replies

  • #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.
 
  • #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:)
I don`t understand what "overlap" means, nor do I understand what "in at most a line segment" means?
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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