# Intro. to Integrals

## Main Question or Discussion Point

Im 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 dont understand what that part in red means??
I dont understand what "overlap" means, nor do I understand what "in at most a line segment" means?

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

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

Finally, the two rectangles $0\le x\le 6$, $0\le y\le 1$, and $7\le x\le 8$, $0\le y\le 1$ do not overlap at all- they have no points in common.

tiny-tim
Homework Helper
Hi wajed!

(btw, it looks better if you type ' rather than  in "don't" etc … the  takes up too much room! )
I dont 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.