1020 billion barrels likely to last?
SOLUTION: Given the yearly consumption in 2000 and the projected
exponential rate of increase in consumption, the anticipated consumption during
the Δtth part of a year (after 2000) is 77e 0.02t Δt billion barrels. The total to be
used during the following N years is therefore 77 e0.02t dt. This integral must
equal 1020 billion barrels.
We must now solve this equation for N. We getEither 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