**1. Introduction.** There are about 4 properties we expect the integral to obey. We expect an integral over a zero width region should vanish

(1)

The fundamental theorem of calculus confirms this. If is the antiderivative for , we see

identically vanishes.

**Additivity.** Likewise, if we have the integral be over and , then

(2)

Again, using the fundamental theorem of calculus we see

and

Setting equals to equals yields (2).

This should make sense, since the intuition behind integration is consider a “*continuous sum*” of thin rectangular strips between and . The width of each strip is (“infinitesimal” wink wink), and we just keep adding more and more strips. In a sense this is just the associativity of addition.

**Linearity of Integrand.** Let , be continuous functions on . Let , be their antiderivatives. Then

(3)

where are constants.

How do we see this? Again, by the fundamental theorem of calculus we have

We see this is precisely the right hand side of (3), after invoking the fundamental theorem of calculus.

Special cases of linearity: taking ; taking and . Also where is a constant, take , .

**Domination Property.** On an interval suppose for each . Then

(4)

The “proof” here is more intuitive: for each . Thus its integral over is positive.

**Inequalities.** Take continuous on . Let for each . Then

(5)

This can follow from the domination property if we take and and observe for each .

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## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.

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