The Definition of the Definite Integral

The Definition of the Definite Integral - The Definition of...

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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 Given a function 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 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 . Let’s work a quick example. This example will use many of the properties and facts from the brief review of summation notation in the Extras chapter. Example 1 Using 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,
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The subintervals are then, As we can see the right endpoint of the i th 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
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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.
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We’ve seen several methods for dealing with the limit in this problem so I’ll leave it to you to verify the results. Wow, that was a lot of work for a fairly simple function. There is a much simpler way
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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The Definition of the Definite Integral - The Definition of...

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