( 4π in.^2 ) (v(t) in./sec)(Δt sec) = 4πv(t) Δt in.^3.The sum of the n amounts of water that drain from the pipe, as n → ∞, is
dt ; the units are cubic inches ( in.^3 ).
Figure N8− 2Example 9 __
Traffic: Total Number of Cars. The density of cars (the number of cars per
mile) on 10 miles of the highway approaching Disney World is equal
approximately to f (x) = 200[ 4 − ln (2x + 3) ], where x is the distance in miles
from the Disney World entrance. Find the total number of cars on this 10-mile
stretch.
SOLUTION: Partition the interval [0, 10] into n equal subintervals each of
width Δx. In each subinterval the number of cars equals approximately the
density of cars f (x) times πx, where f (x) = 200[ 4 − ln (2x + 3) ]. When we add n
of these products we get which is a Riemann Sum. As n → ∞ (or as Δx
→ 0), the Riemann Sum approaches the definite integral
which, using our calculator, is approximately equal to 3118 cars.
Example 10 __
Resource Depletion. In 2000 the yearly world petroleum consumption was
about 77 billion barrels and the yearly exponential rate of increase in use was
2%. How many years after 2000 are the world’s total estimated oil reserves of