Example 36 __
Find L(5), R(5), and T(5) for .
SOLUTION: Noting that for n = 5 subintervals on the interval [1,6] we have Δx
= 1, we make a table of values for :
Then:
NOTE: The calculator finds that is approximately 215.011.
H. AVERAGE VALUE
Average value of a functionIf the function y = f(x) is integrable on the interval a ≤ x ≤ b, then we define the
average value of f from a to b to be
Note that (1) is equivalent to
If f(x) ≥ 0 for all x on [a,b], we can interpret (2) in terms of areas as follows: The
right-hand expression represents the area under the curve of y = f(x), above the x-
axis, and bounded by the vertical lines x = a and x = b. The left-hand expression
of (2) represents the area of a rectangle with the same base (b − a) and with the
average value of f as its height. See Figure N6–17.
CAUTION: The average value of a function is not the same as the average rate
of change. Before answering any question about either of these, be sure to reread
the question carefully to be absolutely certain which is called for.