Note also that the definite integral satisfies certain properties.
For example, linearity. If you have the integral of the sum of
two functions, f and g. Then it's really the sum of the
integrals. Otherwise said, if you add your two
integrands together, and then integrate, you get the same thing as if you
integrate the pieces, and then add them together.
This is true at the level of an individual Riemann sum element.
And so it's true in the limit. Likewise if you multiply and integrand f
by a scalar c then the integral is equal to that constant c times the integral of
f. Again, otherwise said, you can multiply
by a constant and then integrate. Or integrate and then multiply by a
constant. It doesn't matter.
You get to the same place which ever path you take.
Again, the reason why this is true is because it's true at the level of Riemann
sums and hence to a limit. Another important property is that of
additivity. Which states that if you take the
integral of f from a to b and add to it the integral of f from b to c, because
those limits match up you get the integral of f from a to c.
This certainly makes sense at the level of a Riemann sum, you can concatenate
these intervals together. We're going to think of it in terms of
adding paths together, a perspective that makes sense in the context of
orientation. That is, the integral of f from a to b is
minus the integral of f from b to a. Now why does this happen?
Well let's think of the following terms, if we were to move the integral from b to
a over to the left hand side of the equation we would get that the integral
from a to b plus the integral from b to a equals 0.
Why would that have to be true? Well, from additivity the limits match up
and give us the integral from a to a, which clearly must be 0.
That's one way to make sense of this orientation property.
Another way to think about it is that we are adding directed paths together, and
when you add the same path from a to b, with the orientations reversed it's as if
the paths cancel and you wind up getting the integral over a point, which is 0.
The last property we'll discuss is that of dominance.
That states that if f is a non-negative function then the integral of f over an
interval is also non-negative. From that follows a, a slightly less
obvious result. Namely if you have a function g, which is
bigger than f, then g minus f is non-negative.
Which means that the integral of g minus f Is non-negative, which by linearity
means that if g is bigger than f then the integral of g is bigger than the integral
of f. So much for the good news.
The bad news is we can hardly compute anything with this definition.
There are two definite integrals we can compute.
We can compute the integral of a constant by, let's say choosing a uniform
partition and then taking the appropriate limit.
You can see that you get a constant times the width of the interval.
The other integral that we can do is the one that we've done already.
The integral of x dx. If we do that over a general interval
from a to b, then I'll leave it to you to set up the uniform partition, reduce it
to a limit. Then get the answer, which is, as it must
be, 1 half times quantity b squared minus a squared.
That's about it. There's a little bit more that we can do.
For example, if we tried to integrate sine of x or cosine of x.
Not over an arbitrary interval but over a symmetric interval from negative L to L.
Then there are a few things we would observe.
For sine there's a symmetry about the origin which implies that every time you
have a partition element on the right, with say a positive value.
You get a corresponding partition element on the left with the opposite value.
These two will cancel and will give you an integral of 0, because sine of
negative x is minus sine of x. For cosine we can't quite do the same
thing, but we have a symmetry about the y axis.
Which means that every time you have a partition element on the right, it is
balanced by a symmetric partition element with the same value of cosine.
Therefore, we get a doubling. Because cosine of negative x equals
cosine of x, we can reduce this integral to one from 0 to L and double it.
This simple example has a more general pattern.
We say that sine is an odd function and cosine is an even function.
An odd function is one that has this symmetry about the origin or function for
which f of minus x is minus f of x. For such a function, the definite
integral over a symmetric domain from negative L to L is always 0.
Likewise, for an even function, when f of minus x is f of x, then the integral from
negative L to L is twice the integral from 0 to L.
Another way to think about odd and even functions is that the odd ones have an
odd Taylor series and the even ones have an even Taylor series all about 0.
Now in general you're going to have to be careful.
Definite and indefinite integrals are not the same type of object even though they
have similar notation. A definite integral is a number and a
limit of sums. The indefinite integral is an
anti-derivative in a class of functions. We'll soon see what they have in common.
