In this section we use properties of definite integrals to compute and interpret them.
1 Properties of Definite Integrals
We begin with some basic properties of definite integrals- many of which are familiar from our study of derivatives and their basic properties.
Recall that if on an interval then the definite integral, , gives the area under the curve and if on an interval then the definite integral, , gives -1 times the area above the curve. We now consider the situation where the integrand changes sign on the interval . The key to handling this situation is to use the following property.
We apply this proposition in the following examples.
By additivity, we can write Next, we can determine the definite integrals on the right hand side of the above equation by relating them to area. On the interval and so On the interval and so Now, by the additivity of the definite integral,
Since the definite integral is related to area, there are situations where computing these integrals can be facilitated by observing a pattern of symmetry in a region. To aid us in our discussion, we will have to recall the idea of even and odd functions.
The terms even and odd come from the observation that functions of the form are even functions if is even and they are odd functions if is odd. Furthermore, is an even function, while is an odd function. The graph of an even function is symmetric about the -axis and the graph of an odd function is symmetric about the origin. The following proposition describes how these symmetries affect the definite integral.This result is especially important for odd functions, since the calculation of definite integrals in this case becomes trivial. For even functions the advantage of this result less significant. It allows us to work with the number instead of the negative number , thereby simplifying any arithmetic associated with the Fundamental Theorem of Calculus.
Note that in the last two examples, plugging in the value zero gave zero. This is not a
coincidence. An even function always has an odd anti-derivative and odd functions
must pass thru the origin.
Now we look at some examples of odd functions.
In the next example, we combine an even function and an odd function.
2024-09-27 13:56:54