The term on the left-hand side of (37) involves adding the areas of n rectangles constructed using (£,- - r,. J as the base and /((>, + ti_l)/2) as the height. Figure 8 displays this construction. Note that the small area A is approximately equal to the area B. This is especially true if the base of the rectangles is small and if the function f(t) is smooth—that is, does not ; vary heavily in small intervals.

In case the sum of the rectangles fails to approximate the area under the curve, we may be able to correct this by considering a finer partition. As the \tj — j |'s get smaller, the base of the rectangles will get smaller. More rectangles will be available, and the area can be better approximated.

Obviously, the condition that f(t) should be smooth plays an important role during this process. In fact, a very "irregular" path followed by f(t) [■ may be much more difficult to approximate by this method. Using the ter- ; minology discussed before, in order for this method to work, the function : /(f) must be Riemann-integrable.

A counterexample is shown in Figure 9, Here, the function f(t) shows ; steep variations. If such variations do not smooth out as the base of the rectangles gets smaller, the approximation by rectangles may fail.

We have one more comment that will be important in dealing with the : Ito integral later in the text. The rectangles used to approximate the area f(t)

0 0

Post a comment