its equivalent definite integral to calculate the desired quantity.
EXAMPLE 8
Amount of Leaking Water. Water is draining from a cylindrical pipe of radius 2 inches. At t
seconds the water is flowing out with velocity v(t) inches per second. Express the amount of water
that has drained from the pipe in the first 3 minutes as a definite integral in terms of v(t).
SOLUTION: We first express 3 min as 180 sec. We then partition [0,180] into n subintervals
each of length Δt. In Δt sec, approximately v(t) Δt in. of water have drained from the pipe. Since a
typical cross section has area 4π in.^2 (Figure N8–2), in Δt sec the amount that has drained is
(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 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 Δx, 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 integralwhich, 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 1020 billion barrels likely to last?
SOLUTION: Given the yearly consumption in 2000 and the projected exponential rate of increase