Barrons AP Calculus - David Bock

under above the t-axis, and bouned at the left by t = 1 and at the right by t = 2 (the

shaded region in Figure N6–16). Since is strictly decreasing, the area of the inscribed

rectangle (height width 1) is less than ln 2, which, in turn, is less than the area of the

circumscribed rectangle (height 1, width 1). Thus

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

x 1 2 3 4 5 6

f (x) 1206040302420

Then:

NOTE: The calculator finds that is approximately 215.011.

G. AVERAGE VALUE

Average value of a function

If 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.