Barrons AP Calculus - David Bock

FIGURE N6–15c

EXAMPLE 33
Assume the world use of copper has been increasing at a rate given by f (t) = 1.5e0.015t, where t
is measured in years, with t = 0 the beginning of 2000, and f (t) is measured in millions of tons
per year.
(a) What definite integral gives the total amount of copper that was used for the 5-year period
from t = 0 to the beginning of the year 2005?
(b) Write out the terms in the left sum L(5) for the integral in (a). What do the individual terms of
L(5) mean in terms of the world use of copper?
(c) How good an approximation is L(5) for the definite integral in (a)?
SOLUTIONS:
(a)
(b) L(5) = 15e0.015 · 0 + 15e0.015 · 1 + 15e0.015 · 2 + 15e0.015 · 3 + 15e0.015 · 4. The five terms on the
right represent the world’s use of copper for the 5 years from 2000 until 2005.
(c) The answer to (a), using our calculator, is 77.884 million tons. L(5) = 77.301 million tons, so
L(5) underestimates the projected world use of copper during the 5-year period by
approximately 583,000 tons.
Example 32 is an excellent instance of the FTC: if f = F ′ then gives the total change in F
as x varies from a to b.

EXAMPLE 34