The Definition of the Definite Integral
In this section we will formally define the definite
integral and give many of the properties of definite integrals. Let�s start off
with the definition of a definite integral.
Definite Integral
The definite integral is defined to be exactly the limit
and summation that we looked at in the last section to find the net area between
a function and the x-axis. Also note that the notation for the definite
integral is very similar to the notation for an indefinite integral. The reason
for this will be apparent eventually.
There is also a little bit of terminology that we should
get out of the way here. The number �a� that is at the bottom of the
integral sign is called the lower limit of the integral and the number �b�
at the top of the integral sign is called the upper limit of the
integral. Also, despite the fact that a and b were given as an
interval the lower limit does not necessarily need to be smaller than the upper
limit. Collectively we�ll often call a and b the interval of
integration.
ExampleUsing the definition of the definite integral compute the following.
Solution
First, we can�t actually use the definition unless we
determine which points in each interval that well use for
 .
In order to make our life easier we�ll use the right endpoints of each interval.
From the previous section we know that for a general n
the width of each subinterval is,
The subintervals are then,
As we can see the right endpoint of the ith
subinterval is
The summation in the definition of the definite integral is
then,
Now, we are going to have to take a limit of this. That
means that we are going to need to �evaluate� this summation. In other words,
we are going to have to use the formulas given in the
summation notation review to eliminate the actual summation and get a
formula for this for a general n.
To do this we will need to recognize that n is a
constant as far as the summation notation is concerned. As we cycle through the
integers from 1 to n in the summation only i changes and so
anything that isn�t an i will be a constant and can be factored out of
the summation. In particular any n that is in the summation can be
factored out if we need to.
Here is the summation �evaluation�.
We can now compute the definite integral.
We�ve seen several methods for dealing with the limit in
this problem so I�ll leave it to you to verify the results
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that
is continuous on the interval [a,b] we divide the interval into
n subintervals of equal width,
,
and from each interval choose a point,
.
Then the definite integral of f(x) from a to b
is
