Intro to Integrals: Understanding The Addition Property

In summary, the author discusses three basic concepts in Integrals: The Rectangle Property, The Addition Property, and The Comparison Property. The Addition Property states that the area of a region composed of smaller regions that overlap in at most a line segment is the sum of the areas of the smaller regions. "Overlap" means that two regions have points in common, and "in at most a line segment" means that the overlapping area has zero area.
  • #1
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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
 
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  • #2
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
Hi wajed! :smile:

(btw, it looks better if you type ' rather than ` in "don't" etc … the ` takes up too much room! :wink:)
wajed said:
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:
 

1. What is an integral?

An integral is a mathematical concept used to calculate the area under a curve. It is represented by the symbol ∫ and is often referred to as the inverse operation of differentiation. Integrals are used in various fields of science and engineering to model and solve real-world problems.

2. What is the addition property of integrals?

The addition property of integrals states that the integral of the sum of two functions is equal to the sum of their individual integrals. In other words, the integral of a function f(x) + g(x) is equal to the integral of f(x) plus the integral of g(x).

3. How is the addition property of integrals used in real-world applications?

The addition property of integrals is used in various real-world applications, such as calculating the total distance traveled by an object with varying velocity or finding the total amount of work done by a varying force. It is also used in statistics to find the total probability of multiple events occurring.

4. Can the addition property of integrals be applied to any type of function?

Yes, the addition property of integrals can be applied to any type of function, as long as the functions are integrable. This means that the functions must be continuous and have a defined integral within the given limits.

5. How is the addition property of integrals related to the fundamental theorem of calculus?

The addition property of integrals is a consequence of the fundamental theorem of calculus, which states that the integral of a function can be calculated by finding the antiderivative of the integrand and evaluating it at the upper and lower limits of integration. The addition property is a result of the linearity property of integrals, which is a part of the fundamental theorem of calculus.

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