Barrons AP Calculus - David Bock

in consumption, the anticipated consumption during the Δtth part of a year (after 2000) is 77e0.02t

Δt billion barrels. The total to be used during the following N

years is therefore This integral must equal 1020 billion barrels.

We must now solve this equation for N. We get

Either more oil (or alternative sources of energy) must be found, or the world consumption must be

sharply reduced.

D. FTC: DEFINITE INTEGRAL OF A RATE IS NET CHANGE

If f is continuous and then we know from the FTC that

The definite integral of the rate of change of a quantity over an interval is the net change or net
accumulation of the quantity over that interval. Thus, F(b) − F(a) is the net change in F(t) as t varies
from a to b.
We’ve already illustrated this principle many times. Here are more examples.

EXAMPLE 11

Let G(t) be the rate of growth of a population at time t. Then the increase in population between

times t = a and t = b is given by The population may consist of people, deer, fruit flies,

bacteria, and so on.

EXAMPLE 12

Suppose an epidemic is spreading through a city at the rate of f (t) new people per week. Then