1. Introduction. There are about 4 properties we expect the integral to obey. We expect an integral over a zero width region should vanish
The fundamental theorem of calculus confirms this. If is the antiderivative for , we see
Additivity. Likewise, if we have the integral be over and , then
Again, using the fundamental theorem of calculus we see
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
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
The “proof” here is more intuitive: for each . Thus its integral over is positive.
Inequalities. Take continuous on . Let for each . Then
This can follow from the domination property if we take and and observe for each .